Theorem bimsc1 965

Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
2		ancr 321	. . . 4 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑)))
3	1, 2	impbid2 143	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 ∧ 𝜑) ↔ 𝜑))
4	3	bibi2d 232	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜒 ↔ (𝜓 ∧ 𝜑)) ↔ (𝜒 ↔ 𝜑)))
5	4	biimpa 296	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bm1.3ii  4154  bdbm1.3ii  15537