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Theorem imimorbdc 833
Description: Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
imimorbdc (DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))

Proof of Theorem imimorbdc
StepHypRef Expression
1 dfor2dc 832 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ ((𝜓𝜒) → 𝜒)))
21imbi2d 228 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒))))
3 bi2.04 246 . 2 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
42, 3syl6rbbr 197 1 (DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664  DECID wdc 780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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