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Theorem axinfprim 31709
Description: ax-inf 8573 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 9466 . 2 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
2 df-an 385 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝑥) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
32exbii 1814 . . . . . . . . . 10 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
4 exnal 1794 . . . . . . . . . 10 (∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
53, 4bitri 264 . . . . . . . . 9 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
65imbi2i 325 . . . . . . . 8 ((𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
76albii 1787 . . . . . . 7 (∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
87anbi2i 730 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
9 df-an 385 . . . . . 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 325 . . . 4 ((𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1211exbii 1814 . . 3 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
13 df-ex 1745 . . 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 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538  ax-inf 8573
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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