Theorem bimsc1

Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	bimsc1	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 ↔ ( 𝜓 ∧ 𝜑 ) ) ) → ( 𝜒 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜒 ↔ ( 𝜓 ∧ 𝜑 ) ) → ( 𝜒 ↔ ( 𝜓 ∧ 𝜑 ) ) )
2		simpr	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜑 )
3		ancr	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜓 ∧ 𝜑 ) ) )
4	2 3	impbid2	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 ∧ 𝜑 ) ↔ 𝜑 ) )
5	1 4	sylan9bbr	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 ↔ ( 𝜓 ∧ 𝜑 ) ) ) → ( 𝜒 ↔ 𝜑 ) )