Theorem bj-equsal1

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

	Ref	Expression
Hypotheses	bj-equsal1.1	⊢ Ⅎ 𝑥 𝜓
	bj-equsal1.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	bj-equsal1	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-equsal1.1	⊢ Ⅎ 𝑥 𝜓
2		bj-equsal1.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	2	a2i	⊢ ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜓 ) )
4	3	alimi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )
5	1	bj-equsal1ti	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ 𝜓 )
6	4 5	sylib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜓 )