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Theorem wl-ax11-lem7 34264
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 2089 . 2 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2119.28 2160 1 (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 387  wal 1505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747
This theorem is referenced by:  wl-ax11-lem8  34265
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