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Theorem axextprim 32824
Description: ax-ext 2790 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 10001 . 2 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
2 dfbi2 475 . . . . . 6 ((𝑥𝑦𝑥𝑧) ↔ ((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)))
32imbi1i 351 . . . . 5 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ (((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧))
4 impexp 451 . . . . 5 ((((𝑥𝑦𝑥𝑧) ∧ (𝑥𝑧𝑥𝑦)) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
53, 4bitri 276 . . . 4 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
65exbii 1839 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
7 df-ex 1772 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
86, 7bitri 276 . 2 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
91, 8mpbi 231 1 ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-nfc 2960
This theorem is referenced by: (None)
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