Theorem wl-ax11-lem9

Description: The easy part when x coincides with y . (Contributed by Wolf Lammen, 30-Jun-2019)

	Ref	Expression
Assertion	wl-ax11-lem9	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜑 ) )
2	1	dral1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜑 ) )
3	2	aecoms	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜑 ) )
4	3	dral1	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑦 ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )
5	4	aecoms	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜑 ) )