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Theorem axregprim 35540
Description: ax-reg 9628 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
Assertion
Ref Expression
axregprim (𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Proof of Theorem axregprim
StepHypRef Expression
1 axregnd 10638 . 2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2 df-an 395 . . . 4 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
32exbii 1843 . . 3 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
4 exnal 1822 . . 3 (∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
53, 4bitri 274 . 2 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
61, 5sylib 217 1 (𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  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 5296  ax-pr 5425  ax-reg 9628
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  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-un 3951  df-sn 4624  df-pr 4626
This theorem is referenced by: (None)
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