Theorem wl-ax11-lem7 36980

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem7	⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Proof of Theorem wl-ax11-lem7

Step	Hyp	Ref	Expression
1		nfna1 2142	. 2 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2	1	19.28 2214	1 ⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2164

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  wl-ax11-lem8  36981