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Theorem axextdist 32217
Description: ax-ext 2777 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
axextdist ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))

Proof of Theorem axextdist
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfnae 2441 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfnae 2441 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1999 . . 3 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 nfcvf 2965 . . . . . 6 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
54adantr 473 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → 𝑧𝑥)
65nfcrd 2948 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑥)
7 nfcvf 2965 . . . . . 6 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
87adantl 474 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → 𝑧𝑦)
98nfcrd 2948 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑦)
106, 9nfbid 2002 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤𝑥𝑤𝑦))
11 elequ1 2164 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
12 elequ1 2164 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1311, 12bibi12d 337 . . . 4 (𝑤 = 𝑧 → ((𝑤𝑥𝑤𝑦) ↔ (𝑧𝑥𝑧𝑦)))
1413a1i 11 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑤 = 𝑧 → ((𝑤𝑥𝑤𝑦) ↔ (𝑧𝑥𝑧𝑦))))
153, 10, 14cbvald 2414 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤𝑥𝑤𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
16 axext3 2779 . 2 (∀𝑤(𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)
1715, 16syl6bir 246 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651  wnfc 2928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-cleq 2792  df-clel 2795  df-nfc 2930
This theorem is referenced by:  axext4dist  32218
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