Theorem wl-ax11-lem2

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

	Ref	Expression
Assertion	wl-ax11-lem2	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ∀ 𝑥 𝑢 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		sp	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → 𝑢 = 𝑦 )
2		aev	⊢ ( ∀ 𝑥 𝑥 = 𝑢 → ∀ 𝑥 𝑥 = 𝑦 )
3		pm2.21	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑥 𝑥 = 𝑢 ) )
4	2 3	impbid2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑢 ↔ ∀ 𝑥 𝑥 = 𝑦 ) )
5	1 4	anim12i	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ( 𝑢 = 𝑦 ∧ ( ∀ 𝑥 𝑥 = 𝑢 ↔ ∀ 𝑥 𝑥 = 𝑦 ) ) )
6		wl-aleq	⊢ ( ∀ 𝑥 𝑢 = 𝑦 ↔ ( 𝑢 = 𝑦 ∧ ( ∀ 𝑥 𝑥 = 𝑢 ↔ ∀ 𝑥 𝑥 = 𝑦 ) ) )
7	5 6	sylibr	⊢ ( ( ∀ 𝑢 𝑢 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → ∀ 𝑥 𝑢 = 𝑦 )