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Theorem axextprim 35204
Description: ax-ext 2697 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 10585 . 2 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
2 dfbi2 474 . . . . . 6 ((𝑥𝑦𝑥𝑧) ↔ ((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)))
32imbi1i 349 . . . . 5 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ (((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧))
4 impexp 450 . . . . 5 ((((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
53, 4bitri 275 . . . 4 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
65exbii 1842 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
7 df-ex 1774 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
86, 7bitri 275 . 2 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
91, 8mpbi 229 1 ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-clel 2804  df-nfc 2879
This theorem is referenced by: (None)
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