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Theorem axregprim 31708
Description: ax-reg 8538 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 9464 . 2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2 df-an 385 . . . 4 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
32exbii 1814 . . 3 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
4 exnal 1794 . . 3 (∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
53, 4bitri 264 . 2 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
61, 5sylib 208 1 (𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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