Theorem bimsc1 858

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

Assertion

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

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((𝜒 ↔ (𝜓 ∧ 𝜑)) → (𝜒 ↔ (𝜓 ∧ 𝜑)))
2		simpr 490	. . 3 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
3		ancr 556	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑)))
4	2, 3	impbid2 229	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜓 ∧ 𝜑) ↔ 𝜑))
5	1, 4	sylan9bbr 520	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  sepexlem  5254  bj-bm1.3ii  37979