Axiom ax-i12 1453

Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom has been modified from the original ax-12 1457 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
ax-i12	|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

Detailed syntax breakdown of Axiom ax-i12

Step	Hyp	Ref	Expression
1		vz	. . . 4 setvar z
2		vx	. . . 4 setvar x
3	1, 2	weq 1447	. . 3 wff z = x
4	3, 1	wal 1297	. 2 wff A. z z = x
5		vy	. . . . 5 setvar y
6	1, 5	weq 1447	. . . 4 wff z = y
7	6, 1	wal 1297	. . 3 wff A. z z = y
8	2, 5	weq 1447	. . . . 5 wff x = y
9	8, 1	wal 1297	. . . . 5 wff A. z x = y
10	8, 9	wi 4	. . . 4 wff ( x = y -> A. z x = y )
11	10, 1	wal 1297	. . 3 wff A. z ( x = y -> A. z x = y )
12	7, 11	wo 670	. 2 wff ( A. z z = y \/ A. z ( x = y -> A. z x = y ) )
13	4, 12	wo 670	1 wff ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

This axiom is referenced by:  ax-12  1457  ax12or  1458  dveeq2  1754  dveeq2or  1755  dvelimALT  1946  dvelimfv  1947