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Theorem pm4.14dc 821
Description: Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
Assertion
Ref Expression
pm4.14dc (DECID 𝜒 → (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Proof of Theorem pm4.14dc
StepHypRef Expression
1 con34bdc 799 . . 3 (DECID 𝜒 → ((𝜓𝜒) ↔ (¬ 𝜒 → ¬ 𝜓)))
21imbi2d 228 . 2 (DECID 𝜒 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓))))
3 impexp 259 . 2 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
4 impexp 259 . 2 (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
52, 3, 43bitr4g 221 1 (DECID 𝜒 → (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 776
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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