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Theorem pm4.79dc 820
Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
pm4.79dc (DECID 𝜑 → (DECID 𝜓 → (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))))

Proof of Theorem pm4.79dc
StepHypRef Expression
1 id 19 . . . 4 ((𝜓𝜑) → (𝜓𝜑))
2 id 19 . . . 4 ((𝜒𝜑) → (𝜒𝜑))
31, 2jaoa 650 . . 3 (((𝜓𝜑) ∨ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))
4 simplimdc 768 . . . . . 6 (DECID 𝜓 → (¬ (𝜓𝜑) → 𝜓))
5 pm3.3 252 . . . . . 6 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
64, 5syl9 70 . . . . 5 (DECID 𝜓 → (((𝜓𝜒) → 𝜑) → (¬ (𝜓𝜑) → (𝜒𝜑))))
7 dcim 795 . . . . . 6 (DECID 𝜓 → (DECID 𝜑DECID (𝜓𝜑)))
8 pm2.54dc 801 . . . . . 6 (DECID (𝜓𝜑) → ((¬ (𝜓𝜑) → (𝜒𝜑)) → ((𝜓𝜑) ∨ (𝜒𝜑))))
97, 8syl6 33 . . . . 5 (DECID 𝜓 → (DECID 𝜑 → ((¬ (𝜓𝜑) → (𝜒𝜑)) → ((𝜓𝜑) ∨ (𝜒𝜑)))))
106, 9syl5d 66 . . . 4 (DECID 𝜓 → (DECID 𝜑 → (((𝜓𝜒) → 𝜑) → ((𝜓𝜑) ∨ (𝜒𝜑)))))
1110imp 119 . . 3 ((DECID 𝜓DECID 𝜑) → (((𝜓𝜒) → 𝜑) → ((𝜓𝜑) ∨ (𝜒𝜑))))
123, 11impbid2 135 . 2 ((DECID 𝜓DECID 𝜑) → (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑)))
1312expcom 113 1 (DECID 𝜑 → (DECID 𝜓 → (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  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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