Theorem wl-ax11-lem4

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

	Ref	Expression
Assertion	wl-ax11-lem4	⊢ Ⅎ 𝑥 ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		ancom	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) ↔ ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑢 𝑢 = 𝑦 ) )
2		nfna1	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
3		wl-ax11-lem3	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ∀ 𝑢 𝑢 = 𝑦 )
4	2 3	nfan1	⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑢 𝑢 = 𝑦 )
5	1 4	nfxfr	⊢ Ⅎ 𝑥 ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 )