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Theorem imimorbdc 806
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 805 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ ((𝜓𝜒) → 𝜒)))
21imbi2d 223 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒))))
3 bi2.04 241 . 2 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
42, 3syl6rbbr 192 1 (DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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