Theorem bj-equsal

Description: Shorter proof of equsal . (Contributed by BJ, 30-Sep-2018) Proof modification is discouraged to avoid using equsal , but "min */exc equsal" is ok. (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-equsal.1	⊢ Ⅎ 𝑥 𝜓
2		bj-equsal.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	2	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4	1 3	bj-equsal1	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜓 )
5	2	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
6	1 5	bj-equsal2	⊢ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
7	4 6	impbii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )