Modus Ponendo Ponens
Contents
Proof Rule
The modus ponendo ponens is a valid deduction sequent in propositional logic:
If we can conclude $p \implies q$, and we can also conclude $p$, then we may infer $q$.
Thus it provides a means of eliminating a conditional from a sequent.
It can be written:
 $\displaystyle {p \quad p \implies q \over q} \to_e$
Sequent Form
The Modus Ponendo Ponens can be symbolised by the sequent:
 $p \implies q, p \vdash q$
Tableau Form
Let $\phi \implies \psi$ be a propositional formula in a tableau proof whose main connective is the implication operator.
The Modus Ponendo Ponens is invoked for $\phi \implies \psi$ and $\phi$ as follows:
Pool:  The pooled assumptions of $\phi \implies \psi$  
The pooled assumptions of $\phi$  
Formula:  $\psi$  
Description:  Modus Ponendo Ponens  
Depends on:  The line containing the instance of $\phi \implies \psi$  
The line containing the instance of $\phi$  
Abbreviation:  $\text{MPP}$ or $\implies \mathcal E$ 
Variants
The following forms can be used as variants of this theorem:
Variant 1
 $p \vdash \left({p \implies q}\right) \implies q$
Variant 2
 $\vdash p \implies \left({\left({p \implies q}\right) \implies q}\right)$
Variant 3
 $\vdash \left({\left({p \implies q}\right) \land p}\right) \implies q$
Also known as
Modus ponendo ponens is also known as:
 Modus ponens
 The rule of implieselimination
 The rule of material detachment.
Linguistic Note
Modus ponendo ponens is Latin for mode that by affirming, affirms.
Modus ponens means mode that affirms.
Also see
The following are related argument forms:
Technical Note
When invoking Modus Ponendo Ponens in a tableau proof, use the ModusPonens template:
{{ModusPonenslinepoolstatementfirstsecond}}
or:
{{ModusPonenslinepoolstatementfirstsecondcomment}}
where:

line
is the number of the line on the tableau proof where Modus Ponendo Ponens is to be invoked 
pool
is the pool of assumptions (commaseparated list) 
statement
is the statement of logic that is to be displayed in the Formula column, without the$ ... $
delimiters 
first
is the first of the two lines of the tableau proof upon which this line directly depends, the one in the form $p \implies q$ 
second
is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $p$ 
comment
is the (optional) comment that is to be displayed in the Notes column.
Sources
 Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964)... (previous)... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
 E.J. Lemmon: Beginning Logic (1965)... (previous)... (next): $\S 1.2$: Conditionals and Negation
 A. G. Howson: A Handbook of Terms used in Algebra and Analysis (1972)... (previous)... (next): $\S 1$: Some mathematical language: Axioms
 Irving M. Copi: Symbolic Logic (4th ed., 1973)... (previous)... (next): $2.3$: Argument Forms and Truth Tables
 Irving M. Copi: Symbolic Logic (4th ed., 1973)... (previous)... (next): $3.1$: Formal Proof of Validity
 D.J. O'Connor and Betty Powell: Elementary Logic (1980)... (previous)... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $1$
 P.M. Cohn: Algebra Volume 1 (2nd ed., 1982)... (previous)... (next): $\S 1.1$: The need for logic
 H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996)... (previous)... (next): $\S 1.12$: Valid Arguments
 Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000)... (previous)... (next): $\S 1.2.1$: Rules for natural deduction