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Theorem imimorbdc 864
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 863 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ ((𝜓𝜒) → 𝜒)))
21imbi2d 229 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒))))
3 bi2.04 247 . 2 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
42, 3syl6rbbr 198 1 (DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 680  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by: (None)
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