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Theorem axinfprim 33167
 Description: ax-inf 9139 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 10071 . 2 𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
2 df-an 400 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝑥) ↔ ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
32exbii 1849 . . . . . . . . . 10 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥))
4 exnal 1828 . . . . . . . . . 10 (∃𝑧 ¬ (𝑦𝑧 → ¬ 𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
53, 4bitri 278 . . . . . . . . 9 (∃𝑧(𝑦𝑧𝑧𝑥) ↔ ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))
65imbi2i 339 . . . . . . . 8 ((𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
76albii 1821 . . . . . . 7 (∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))
87anbi2i 625 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
9 df-an 400 . . . . . 6 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
108, 9bitri 278 . . . . 5 ((𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))) ↔ ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
1110imbi2i 339 . . . 4 ((𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1211exbii 1849 . . 3 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
13 df-ex 1782 . . 3 (∃𝑥(𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
1412, 13bitri 278 . 2 (∃𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))) ↔ ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥)))))
151, 14mpbi 233 1 ¬ ∀𝑥 ¬ (𝑦𝑧 → ¬ (𝑦𝑥 → ¬ ∀𝑦(𝑦𝑥 → ¬ ∀𝑧(𝑦𝑧 → ¬ 𝑧𝑥))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-reg 9094  ax-inf 9139 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-nul 4228  df-sn 4526  df-pr 4528 This theorem is referenced by: (None)
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