Axiom ax-10 2140

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 2139 ("o" for "old") and was replaced with this shorter ax-10 2140 in May 2008. The old axiom is proved from this one as Theorem ax10o 1952. Conversely, this axiom is proved from ax-10o 2139 as Theorem ax10from10o 2177.

This axiom was proved redundant in July 2015. See Theorem ax10 1944.

This axiom is obsolete and should no longer be used. It is proved above as Theorem ax10 1944. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-10	⊢ (∀x x = y → ∀y y = x)

Detailed syntax breakdown of Axiom ax-10

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar x
2		vy	. . . 4 setvar y
3	1, 2	weq 1643	. . 3 wff x = y
4	3, 1	wal 1540	. 2 wff ∀x x = y
5	2, 1	weq 1643	. . 3 wff y = x
6	5, 2	wal 1540	. 2 wff ∀y y = x
7	4, 6	wi 4	1 wff (∀x x = y → ∀y y = x)

This axiom is referenced by:  ax10o-o  2203