Theorem bimsc1 976

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		simpr 476	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
2		ancr 570	. . . 4 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑)))
3	1, 2	impbid2 215	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 ∧ 𝜑) ↔ 𝜑))
4	3	bibi2d 331	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜒 ↔ (𝜓 ∧ 𝜑)) ↔ (𝜒 ↔ 𝜑)))
5	4	biimpa 500	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by:  bm1.3ii  4712