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Axiom ax-c14 33069
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 1793; see theorem axc14 2264. Alternately, ax-5 1793 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1793. We retain ax-c14 33069 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1793, 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This axiom is obsolete and should no longer be used.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 2264. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c14 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Detailed syntax breakdown of Axiom ax-c14
StepHypRef Expression
1 vz . . . . 5 setvar 𝑧
2 vx . . . . 5 setvar 𝑥
31, 2weq 1824 . . . 4 wff 𝑧 = 𝑥
43, 1wal 1472 . . 3 wff 𝑧 𝑧 = 𝑥
54wn 3 . 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6 vy . . . . . 6 setvar 𝑦
71, 6weq 1824 . . . . 5 wff 𝑧 = 𝑦
87, 1wal 1472 . . . 4 wff 𝑧 𝑧 = 𝑦
98wn 3 . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
102, 6wel 1939 . . . 4 wff 𝑥𝑦
1110, 1wal 1472 . . . 4 wff 𝑧 𝑥𝑦
1210, 11wi 4 . . 3 wff (𝑥𝑦 → ∀𝑧 𝑥𝑦)
139, 12wi 4 . 2 wff (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
145, 13wi 4 1 wff (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  ax5el  33115  ax12el  33120
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