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Theorem bimsc1 905
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 108 . . . 4 ((𝜓𝜑) → 𝜑)
2 ancr 314 . . . 4 ((𝜑𝜓) → (𝜑 → (𝜓𝜑)))
31, 2impbid2 141 . . 3 ((𝜑𝜓) → ((𝜓𝜑) ↔ 𝜑))
43bibi2d 230 . 2 ((𝜑𝜓) → ((𝜒 ↔ (𝜓𝜑)) ↔ (𝜒𝜑)))
54biimpa 290 1 (((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bm1.3ii  3907  bdbm1.3ii  10840
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