Axiom ax-12o 2142

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This axiom is obsolete and should no longer be used. It is proved above as Theorem ax12o 1934. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-12o	⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))

Detailed syntax breakdown of Axiom ax-12o

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar z
2		vx	. . . . 5 setvar x
3	1, 2	weq 1643	. . . 4 wff z = x
4	3, 1	wal 1540	. . 3 wff ∀z z = x
5	4	wn 3	. 2 wff ¬ ∀z z = x
6		vy	. . . . . 6 setvar y
7	1, 6	weq 1643	. . . . 5 wff z = y
8	7, 1	wal 1540	. . . 4 wff ∀z z = y
9	8	wn 3	. . 3 wff ¬ ∀z z = y
10	2, 6	weq 1643	. . . 4 wff x = y
11	10, 1	wal 1540	. . . 4 wff ∀z x = y
12	10, 11	wi 4	. . 3 wff (x = y → ∀z x = y)
13	9, 12	wi 4	. 2 wff (¬ ∀z z = y → (x = y → ∀z x = y))
14	5, 13	wi 4	1 wff (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))

This axiom is referenced by:  hbae-o  2153  ax12from12o  2156  equid1  2158  hbequid  2160  equid1ALT  2176  dvelimf-o  2180  ax17eq  2183