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Theorem wl-ax11-lem7 37111
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem7 (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Proof of Theorem wl-ax11-lem7
StepHypRef Expression
1 nfna1 2141 . 2 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2119.28 2216 1 (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778
This theorem is referenced by:  wl-ax11-lem8  37112
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