Axiom ax-c9 38871

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This axiom is obsolete and should no longer be used.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 2384. (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 1959	. . . 4 wff 𝑧 = 𝑥
4	3, 1	wal 1534	. . 3 wff ∀𝑧 𝑧 = 𝑥
5	4	wn 3	. 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6		vy	. . . . . 6 setvar 𝑦
7	1, 6	weq 1959	. . . . 5 wff 𝑧 = 𝑦
8	7, 1	wal 1534	. . . 4 wff ∀𝑧 𝑧 = 𝑦
9	8	wn 3	. . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
10	2, 6	weq 1959	. . . 4 wff 𝑥 = 𝑦
11	10, 1	wal 1534	. . . 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  38880  hbae-o  38884  ax13fromc9  38887  hbequid  38890  equid1ALT  38906  dvelimf-o  38910  ax5eq  38913