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Theorem axinfprim 35686
Description: ax-inf 9676 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 10644 . 2 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
2 df-an 396 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝑥) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
32exbii 1845 . . . . . . . . . 10 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
4 exnal 1824 . . . . . . . . . 10 (∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
53, 4bitri 275 . . . . . . . . 9 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
65imbi2i 336 . . . . . . . 8 ((𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
76albii 1816 . . . . . . 7 (∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
87anbi2i 623 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
9 df-an 396 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
108, 9bitri 275 . . . . 5 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
1110imbi2i 336 . . . 4 ((𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1211exbii 1845 . . 3 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
13 df-ex 1777 . . 3 (∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1412, 13bitri 275 . 2 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
151, 14mpbi 230 1 ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630  ax-inf 9676
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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