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Theorem bimsc1 930
Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
bimsc1 (((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))

Proof of Theorem bimsc1
StepHypRef Expression
1 simpr 109 . . . 4 ((𝜓𝜑) → 𝜑)
2 ancr 317 . . . 4 ((𝜑𝜓) → (𝜑 → (𝜓𝜑)))
31, 2impbid2 142 . . 3 ((𝜑𝜓) → ((𝜓𝜑) ↔ 𝜑))
43bibi2d 231 . 2 ((𝜑𝜓) → ((𝜒 ↔ (𝜓𝜑)) ↔ (𝜒𝜑)))
54biimpa 292 1 (((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bm1.3ii  4017  bdbm1.3ii  12923
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