Theorem wl-ax11-lem6

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

	Ref	Expression
Assertion	wl-ax11-lem6	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-wl-11v	⊢ ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 → ∀ 𝑥 ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 )
2		ax-wl-11v	⊢ ( ∀ 𝑥 ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 → ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 )
3	1 2	impbii	⊢ ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 )
4		nfna1	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
5		wl-ax11-lem3	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 ∀ 𝑢 𝑢 = 𝑦 )
6	4 5	nfan1	⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑢 𝑢 = 𝑦 )
7		wl-ax11-lem5	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → ( ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 𝜑 ) )
8	7	adantl	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑢 𝑢 = 𝑦 ) → ( ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 𝜑 ) )
9	6 8	albid	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑢 𝑢 = 𝑦 ) → ( ∀ 𝑥 ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
10	9	ancoms	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑥 ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
11	3 10	syl5bb	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( ∀ 𝑢 ∀ 𝑥 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )