Equality is Transitive

Theorem

Equality is transitive:

$\forall a, b, c: \paren {a = b} \land \paren {b = c} \implies a = c$

$\ds a$

$=$

$\ds b$

$\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 \ \ $

$\ds \map P a$

$\iff$

$\ds \map P c$

$\ds \vdash \ \ $

$\ds a$

$=$

$\ds c$

Leibniz's law

$\blacksquare$

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: $\text{(c)}$