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Theorem axpowprim 35705
Description: ax-pow 5364 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 10642 . . 3 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
2 df-ex 1779 . . . . . . . . 9 (∃𝑧 𝑥𝑦 ↔ ¬ ∀𝑧 ¬ 𝑥𝑦)
32imbi1i 349 . . . . . . . 8 ((∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ (¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
43albii 1818 . . . . . . 7 (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ ∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
54imbi1i 349 . . . . . 6 ((∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ (∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
65albii 1818 . . . . 5 (∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
76exbii 1847 . . . 4 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
8 df-ex 1779 . . . 4 (∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
97, 8bitri 275 . . 3 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
101, 9sylib 218 . 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 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-pw 4601  df-sn 4626  df-pr 4628
This theorem is referenced by: (None)
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