Axiom ax-c9 35419

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This axiom is obsolete and should no longer be used.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 2310. (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 1922	. . . 4 wff 𝑧 = 𝑥
4	3, 1	wal 1505	. . 3 wff ∀𝑧 𝑧 = 𝑥
5	4	wn 3	. 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6		vy	. . . . . 6 setvar 𝑦
7	1, 6	weq 1922	. . . . 5 wff 𝑧 = 𝑦
8	7, 1	wal 1505	. . . 4 wff ∀𝑧 𝑧 = 𝑦
9	8	wn 3	. . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
10	2, 6	weq 1922	. . . 4 wff 𝑥 = 𝑦
11	10, 1	wal 1505	. . . 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:  equid1  35428  hbae-o  35432  ax13fromc9  35435  hbequid  35438  equid1ALT  35454  dvelimf-o  35458  ax5eq  35461