Axiom ax-15 2143

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-17 1616; see Theorem ax15 2021. Alternately, ax-17 1616 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 1616. We retain ax-15 2143 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 1616, 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This axiom is obsolete and should no longer be used.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 ax15 2021. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-15	⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))

Detailed syntax breakdown of Axiom ax-15

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar z
2		vx	. . . . 5 setvar x
3	1, 2	weq 1643	. . . 4 wff z = x
4	3, 1	wal 1540	. . 3 wff ∀z z = x
5	4	wn 3	. 2 wff ¬ ∀z z = x
6		vy	. . . . . 6 setvar y
7	1, 6	weq 1643	. . . . 5 wff z = y
8	7, 1	wal 1540	. . . 4 wff ∀z z = y
9	8	wn 3	. . 3 wff ¬ ∀z z = y
10	2, 6	wel 1711	. . . 4 wff x ∈ y
11	10, 1	wal 1540	. . . 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:  ax17el  2189  ax11el  2194