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Theorem axrepprim 33159
 Description: ax-rep 5156 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
Assertion
Ref Expression
axrepprim ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))

Proof of Theorem axrepprim
StepHypRef Expression
1 axrepnd 10054 . 2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))
2 df-ex 1782 . . . . 5 (∃𝑦𝑧(𝜑𝑧 = 𝑦) ↔ ¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦))
3 df-an 400 . . . . . . . . . 10 ((∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
43exbii 1849 . . . . . . . . 9 (∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥 ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
5 exnal 1828 . . . . . . . . 9 (∃𝑥 ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
64, 5bitri 278 . . . . . . . 8 (∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
76bibi2i 341 . . . . . . 7 ((∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (∀𝑦 𝑧𝑥 ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)))
8 dfbi1 216 . . . . . . 7 ((∀𝑦 𝑧𝑥 ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) ↔ ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
97, 8bitri 278 . . . . . 6 ((∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
109albii 1821 . . . . 5 (∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
112, 10imbi12i 354 . . . 4 ((∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥))))
1211exbii 1849 . . 3 (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥))))
13 df-ex 1782 . . 3 (∃𝑥(¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥))) ↔ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥))))
1412, 13bitri 278 . 2 (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥))))
151, 14mpbi 233 1 ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-reg 9089 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-nul 4226  df-sn 4523  df-pr 4525 This theorem is referenced by: (None)
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