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Theorem pm4.14dc 787
 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 765 . . 3 (DECID 𝜒 → ((𝜓𝜒) ↔ (¬ 𝜒 → ¬ 𝜓)))
21imbi2d 219 . 2 (DECID 𝜒 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓))))
3 impexp 250 . 2 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
4 impexp 250 . 2 (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
52, 3, 43bitr4g 212 1 (DECID 𝜒 → (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  pm3.37dc  788
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