Theorem bj-equsal2

Description: One direction of equsal . (Contributed by BJ, 30-Sep-2018)

	Ref	Expression
Hypotheses	bj-equsal2.1	⊢ Ⅎ 𝑥 𝜑
	bj-equsal2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	bj-equsal2	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-equsal2.1	⊢ Ⅎ 𝑥 𝜑
2		bj-equsal2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1	bj-equsal1ti	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 )
4	2	a2i	⊢ ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜓 ) )
5	4	alimi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )
6	3 5	sylbir	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )