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Theorem axinfprim 31288
Description: ax-inf 8479 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
Assertion
Ref Expression
axinfprim ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))

Proof of Theorem axinfprim
StepHypRef Expression
1 axinfnd 9372 . 2 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
2 df-an 386 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝑥) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
32exbii 1771 . . . . . . . . . 10 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
4 exnal 1751 . . . . . . . . . 10 (∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
53, 4bitri 264 . . . . . . . . 9 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
65imbi2i 326 . . . . . . . 8 ((𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
76albii 1744 . . . . . . 7 (∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
87anbi2i 729 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
9 df-an 386 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
108, 9bitri 264 . . . . 5 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
1110imbi2i 326 . . . 4 ((𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1211exbii 1771 . . 3 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
13 df-ex 1702 . . 3 (∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1412, 13bitri 264 . 2 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
151, 14mpbi 220 1 ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-reg 8441  ax-inf 8479
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by: (None)
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