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Theorem axrepprim 34085
Description: ax-rep 5240 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 10488 . 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 397 . . . . . . . . . 10 ((∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
43exbii 1850 . . . . . . . . 9 (∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥 ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
5 exnal 1829 . . . . . . . . 9 (∃𝑥 ¬ (∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
64, 5bitri 274 . . . . . . . 8 (∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑))
76bibi2i 337 . . . . . . 7 ((∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (∀𝑦 𝑧𝑥 ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)))
8 dfbi1 212 . . . . . . 7 ((∀𝑦 𝑧𝑥 ↔ ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) ↔ ¬ ((∀𝑦 𝑧𝑥 → ¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧𝑥)))
97, 8bitri 274 . . . . . 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 350 . . . 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 1850 . . 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 274 . 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 229 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 205  wa 396  wal 1539  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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-pr 5382  ax-reg 9486
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-v 3445  df-un 3913  df-sn 4585  df-pr 4587
This theorem is referenced by: (None)
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