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Theorem axc14 2400
Description: Axiom ax-c14 35478 is redundant if we assume ax-5 1869. 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 2399 and ax-5 1869. 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 2080 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦)
2 dveel2 2399 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤𝑦 → ∀𝑧 𝑤𝑦))
31, 2hbim1 2231 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦))
4 elequ1 2057 . . . . 5 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
54imbi2d 333 . . . 4 (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑤𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
63, 5dvelim 2387 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦)))
7 nfa1 2088 . . . . 5 𝑧𝑧 𝑧 = 𝑦
87nfn 1819 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9819.21 2136 . . 3 (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦))
106, 9syl6ib 243 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦𝑥𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥𝑦)))
1110pm2.86d 108 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747
This theorem is referenced by: (None)
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