Theorem wl-ax11-lem5

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

	Ref	Expression
Assertion	wl-ax11-lem5	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → ( ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	⊢ ( 𝑢 = 𝑦 → ( [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
2	1	sps	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → ( [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
3	2	dral1	⊢ ( ∀ 𝑢 𝑢 = 𝑦 → ( ∀ 𝑢 [ 𝑢 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 𝜑 ) )