Equality is Symmetric

Theorem

Equality is symmetric.


That is:

$\forall a, b: a = b \implies b = a$


Proof

\(\ds a\) \(=\) \(\ds b\)
\(\ds \vdash \ \ \) \(\ds \map P a\) \(\iff\) \(\ds \map P b\) Leibniz's Law
\(\ds \vdash \ \ \) \(\ds \map P b\) \(\iff\) \(\ds \map P a\) Biconditional is Commutative
\(\ds \vdash \ \ \) \(\ds b\) \(=\) \(\ds a\) Leibniz's Law

$\blacksquare$


Also see


Sources

  • 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.): Chapter $3$
  • 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: $\text{(b)}$
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