HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Axiom ax-10 1625
Description: Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 1810 ("o" for "old") and was replaced with this shorter ax-10 1625 in May 2008. The old axiom is proved from this one as theorem ax10o 1809. Conversely, this axiom is proved from ax-10o 1810 as theorem ax10 1811.

Assertion
Ref Expression
ax-10 |- (A.x x = y -> A.y y = x)

Detailed syntax breakdown of Axiom ax-10
StepHypRef Expression
1 vx . . . . 5 set x
21cv 1614 . . . 4 class x
3 vy . . . . 5 set y
43cv 1614 . . . 4 class y
52, 4wceq 1615 . . 3 wff x = y
65, 1wal 1613 . 2 wff A.x x = y
74, 2wceq 1615 . . 3 wff y = x
87, 3wal 1613 . 2 wff A.y y = x
96, 8wi 3 1 wff (A.x x = y -> A.y y = x)
Colors of variables: wff set class
This axiom is referenced by:  ax10o 1809  alequcom 1812
Copyright terms: Public domain