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Theorem axextprim 30675
Description: ax-ext 2494 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 9168 . 2 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
2 dfbi2 657 . . . . . 6 ((𝑥𝑦𝑥𝑧) ↔ ((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)))
32imbi1i 337 . . . . 5 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ (((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧))
4 impexp 460 . . . . 5 ((((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
53, 4bitri 262 . . . 4 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
65exbii 1752 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
7 df-ex 1695 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
86, 7bitri 262 . 2 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
91, 8mpbi 218 1 ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-cleq 2507  df-clel 2510  df-nfc 2644
This theorem is referenced by: (None)
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