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Theorem axextprim 31906
Description: ax-ext 2740 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 9625 . 2 𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
2 dfbi2 663 . . . . . 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 264 . . . 4 (((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
65exbii 1923 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
7 df-ex 1854 . . 3 (∃𝑥((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
86, 7bitri 264 . 2 (∃𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧)))
91, 8mpbi 220 1 ¬ ∀𝑥 ¬ ((𝑥𝑦𝑥𝑧) → ((𝑥𝑧𝑥𝑦) → 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-cleq 2753  df-clel 2756  df-nfc 2891
This theorem is referenced by: (None)
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