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Theorem axc14 2255
Description: Axiom ax-c14 32696 is redundant if we assume ax-5 1789. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that 𝑤 is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2254 and ax-5 1789. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

Assertion
Ref Expression
axc14 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Proof of Theorem axc14
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 hbn1 1966 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦)
2 dveel2 2254 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤𝑦 → ∀𝑧 𝑤𝑦))
31, 2hbim1 2054 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦))
4 elequ1 1944 . . . . 5 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
54imbi2d 325 . . . 4 (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
63, 5dvelim 2220 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
7 nfa1 2032 . . . . 5 𝑧𝑧 𝑧 = 𝑦
87nfn 2036 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9819.21 2040 . . 3 (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦))
106, 9syl6ib 236 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦)))
1110pm2.86d 103 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693  df-nf 1697
This theorem is referenced by: (None)
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