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Theorem axpowprim 31286
Description: ax-pow 4803 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 9367 . . 3 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
2 df-ex 1702 . . . . . . . . 9 (∃𝑧 𝑥𝑦 ↔ ¬ ∀𝑧 ¬ 𝑥𝑦)
32imbi1i 339 . . . . . . . 8 ((∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ (¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
43albii 1744 . . . . . . 7 (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ ∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
54imbi1i 339 . . . . . 6 ((∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ (∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
65albii 1744 . . . . 5 (∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
76exbii 1771 . . . 4 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
8 df-ex 1702 . . . 4 (∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
97, 8bitri 264 . . 3 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
101, 9sylib 208 . 2 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
1110con4i 113 1 (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151
This theorem is referenced by: (None)
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