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Theorem imimorbdc 896
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 bi2.04 248 . 2 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
2 dfor2dc 895 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ ((𝜓𝜒) → 𝜒)))
32imbi2d 230 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒))))
41, 3bitr4id 199 1 (DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by: (None)
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