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Theorem axextprim 30832
Description: ax-ext 2590 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
Assertion
Ref Expression
axextprim ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))

Proof of Theorem axextprim
StepHypRef Expression
1 axextnd 9292 . 2 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
2 dfbi2 658 . . . . . 6 ((𝑥𝑦𝑥𝑧) ↔ ((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)))
32imbi1i 338 . . . . 5 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ (((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧))
4 impexp 461 . . . . 5 ((((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
53, 4bitri 263 . . . 4 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
65exbii 1764 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
7 df-ex 1696 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
86, 7bitri 263 . 2 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
91, 8mpbi 219 1 ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740
This theorem is referenced by: (None)
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