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Theorem axregprim 32183
Description: ax-reg 8788 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 9763 . 2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2 df-an 387 . . . 4 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
32exbii 1892 . . 3 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
4 exnal 1870 . . 3 (∃𝑥 ¬ (𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
53, 4bitri 267 . 2 (∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)) ↔ ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
61, 5sylib 210 1 (𝑥𝑦 → ¬ ∀𝑥(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wal 1599  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-reg 8788
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-un 3797  df-nul 4142  df-sn 4399  df-pr 4401
This theorem is referenced by: (None)
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