Axiom ax-c9 40040

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This axiom is obsolete and should no longer be used.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 axc9 2411. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c9	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Detailed syntax breakdown of Axiom ax-c9

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar 𝑧
2		vx	. . . . 5 setvar 𝑥
3	1, 2	weq 1994	. . . 4 wff 𝑧 = 𝑥
4	3, 1	wal 1567	. . 3 wff ∀𝑧 𝑧 = 𝑥
5	4	wn 3	. 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6		vy	. . . . . 6 setvar 𝑦
7	1, 6	weq 1994	. . . . 5 wff 𝑧 = 𝑦
8	7, 1	wal 1567	. . . 4 wff ∀𝑧 𝑧 = 𝑦
9	8	wn 3	. . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
10	2, 6	weq 1994	. . . 4 wff 𝑥 = 𝑦
11	10, 1	wal 1567	. . . 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 used by:  equid1  40049  hbae-o  40053  ax13fromc9  40056  hbequid  40059  equid1ALT  40075  dvelimf-o  40079  ax5eq  40082