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Theorem axregprim 35685
Description: ax-reg 9630 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 10642 . 2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2 df-an 396 . . . 4 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
32exbii 1845 . . 3 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
4 exnal 1824 . . 3 (∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
53, 4bitri 275 . 2 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
61, 5sylib 218 1 (𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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