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Theorem axc14 2359
Description: Axiom ax-c14 32990 is redundant if we assume ax-5 1826. 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 2358 and ax-5 1826. 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 2006 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦)
2 dveel2 2358 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤𝑦 → ∀𝑧 𝑤𝑦))
31, 2hbim1 2109 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦))
4 elequ1 1983 . . . . 5 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
54imbi2d 328 . . . 4 (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
63, 5dvelim 2324 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
7 nfa1 2014 . . . . 5 𝑧𝑧 𝑧 = 𝑦
87nfn 1767 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9819.21 2061 . . 3 (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦))
106, 9syl6ib 239 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦)))
1110pm2.86d 104 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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