Axiom ax-c14 37756

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 1913; see Theorem axc14 2462. Alternately, ax-5 1913 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1913. We retain ax-c14 37756 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1913, 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This axiom is obsolete and should no longer be used.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 2462. (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 1966	. . . 4 wff 𝑧 = 𝑥
4	3, 1	wal 1539	. . 3 wff ∀𝑧 𝑧 = 𝑥
5	4	wn 3	. 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6		vy	. . . . . 6 setvar 𝑦
7	1, 6	weq 1966	. . . . 5 wff 𝑧 = 𝑦
8	7, 1	wal 1539	. . . 4 wff ∀𝑧 𝑧 = 𝑦
9	8	wn 3	. . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
10	2, 6	wel 2107	. . . 4 wff 𝑥 ∈ 𝑦
11	10, 1	wal 1539	. . . 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  37802  ax12el  37807