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Theorem axextdist 31427
Description: ax-ext 2601 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 2317 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 nfnae 2317 . . . 4 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1825 . . 3 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 nfcvf 2784 . . . . . 6 (¬ ∀𝑧 𝑧 = 𝑥𝑧𝑥)
54adantr 481 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → 𝑧𝑥)
65nfcrd 2767 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑥)
7 nfcvf 2784 . . . . . 6 (¬ ∀𝑧 𝑧 = 𝑦𝑧𝑦)
87adantl 482 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → 𝑧𝑦)
98nfcrd 2767 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤𝑦)
106, 9nfbid 1829 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤𝑥𝑤𝑦))
11 elequ1 1994 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
12 elequ1 1994 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1311, 12bibi12d 335 . . . 4 (𝑤 = 𝑧 → ((𝑤𝑥𝑤𝑦) ↔ (𝑧𝑥𝑧𝑦)))
1413a1i 11 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑤 = 𝑧 → ((𝑤𝑥𝑤𝑦) ↔ (𝑧𝑥𝑧𝑦))))
153, 10, 14cbvald 2276 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤𝑥𝑤𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
16 axext3 2603 . 2 (∀𝑤(𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)
1715, 16syl6bir 244 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-cleq 2614  df-clel 2617  df-nfc 2750
This theorem is referenced by:  axext4dist  31428
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