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Theorem axpowprim 32178
Description: ax-pow 5077 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
Assertion
Ref Expression
axpowprim (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) → 𝑥 = 𝑦)

Proof of Theorem axpowprim
StepHypRef Expression
1 axpownd 9758 . . 3 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
2 df-ex 1824 . . . . . . . . 9 (∃𝑧 𝑥𝑦 ↔ ¬ ∀𝑧 ¬ 𝑥𝑦)
32imbi1i 341 . . . . . . . 8 ((∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ (¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
43albii 1863 . . . . . . 7 (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ ∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
54imbi1i 341 . . . . . 6 ((∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ (∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
65albii 1863 . . . . 5 (∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
76exbii 1892 . . . 4 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
8 df-ex 1824 . . . 4 (∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
97, 8bitri 267 . . 3 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
101, 9sylib 210 . 2 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
1110con4i 114 1 (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1599  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-reg 8786
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-pw 4381  df-sn 4399  df-pr 4401
This theorem is referenced by: (None)
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