Theorem stdpc6 2025

Description: One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2248.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)

Assertion

Ref	Expression
stdpc6	⊢ ∀𝑥 𝑥 = 𝑥

Proof of Theorem stdpc6

Step	Hyp	Ref	Expression
1		equid 2009	. 2 ⊢ 𝑥 = 𝑥
2	1	ax-gen 1792	1 ⊢ ∀𝑥 𝑥 = 𝑥

Syntax hints:  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by: (None)