Equality is Transitive
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Theorem
Equality is transitive:
- $\forall a, b, c: \paren {a = b} \land \paren {b = c} \implies a = c$
Proof
| \(\ds a\) | \(=\) | \(\ds b\) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds \map P a\) | \(\iff\) | \(\ds \map P b\) | Leibniz's law | ||||||||||
| \(\ds b\) | \(=\) | \(\ds c\) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds \map P b\) | \(\iff\) | \(\ds \map P c\) | Leibniz's law | ||||||||||
| \(\ds \vdash \ \ \) | \(\ds \map P a\) | \(\iff\) | \(\ds \map P c\) | Biconditional is Transitive | ||||||||||
| \(\ds \vdash \ \ \) | \(\ds a\) | \(=\) | \(\ds c\) | Leibniz's law |
$\blacksquare$
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 3.2$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Equality: $\text{(c)}$