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Theorem axinfprim 35428
Description: ax-inf 9663 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 10631 . 2 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
2 df-an 395 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝑥) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
32exbii 1842 . . . . . . . . . 10 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
4 exnal 1821 . . . . . . . . . 10 (∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
53, 4bitri 274 . . . . . . . . 9 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
65imbi2i 335 . . . . . . . 8 ((𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
76albii 1813 . . . . . . 7 (∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
87anbi2i 621 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
9 df-an 395 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
108, 9bitri 274 . . . . 5 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
1110imbi2i 335 . . . 4 ((𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1211exbii 1842 . . 3 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
13 df-ex 1774 . . 3 (∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1412, 13bitri 274 . 2 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
151, 14mpbi 229 1 ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365  ax-ext 2696  ax-sep 5300  ax-pr 5429  ax-reg 9617  ax-inf 9663
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-v 3463  df-un 3949  df-sn 4631  df-pr 4633
This theorem is referenced by: (None)
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