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Theorem axpowprim 35526
Description: ax-pow 5369 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 10644 . . 3 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
2 df-ex 1775 . . . . . . . . 9 (∃𝑧 𝑥𝑦 ↔ ¬ ∀𝑧 ¬ 𝑥𝑦)
32imbi1i 348 . . . . . . . 8 ((∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ (¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
43albii 1814 . . . . . . 7 (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) ↔ ∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧))
54imbi1i 348 . . . . . 6 ((∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ (∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
65albii 1814 . . . . 5 (∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
76exbii 1843 . . . 4 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
8 df-ex 1775 . . . 4 (∃𝑥𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
97, 8bitri 274 . . 3 (∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
101, 9sylib 217 . 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 1532  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-reg 9635
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634  df-pr 4636
This theorem is referenced by: (None)
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