Axiom ax-c14 34692

Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-5 1990; see theorem axc14 2518. Alternately, ax-5 1990 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1990. We retain ax-c14 34692 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1990, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc14 2518. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c14	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))

Detailed syntax breakdown of Axiom ax-c14

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar 𝑧
2		vx	. . . . 5 setvar 𝑥
3	1, 2	weq 2042	. . . 4 wff 𝑧 = 𝑥
4	3, 1	wal 1628	. . 3 wff ∀𝑧 𝑧 = 𝑥
5	4	wn 3	. 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6		vy	. . . . . 6 setvar 𝑦
7	1, 6	weq 2042	. . . . 5 wff 𝑧 = 𝑦
8	7, 1	wal 1628	. . . 4 wff ∀𝑧 𝑧 = 𝑦
9	8	wn 3	. . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
10	2, 6	wel 2145	. . . 4 wff 𝑥 ∈ 𝑦
11	10, 1	wal 1628	. . . 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:  ax5el  34738  ax12el  34743