Theorem wl-lem-exsb

Description: This theorem provides a basic working step in proving theorems about E* or E! . (Contributed by Wolf Lammen, 3-Oct-2019)

	Ref	Expression
Assertion	wl-lem-exsb	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax12v2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
3	2	com12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
4	1 3	impbid	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )