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Theorem ifpim1g 37348
Description: Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpim1g ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))

Proof of Theorem ifpim1g
StepHypRef Expression
1 ifpim123g 37347 . 2 ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((¬ 𝜓𝜑) ∨ (𝜃𝜃)))))
2 id 22 . . . . . 6 (𝜒𝜒)
32olci 406 . . . . 5 ((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒))
43biantrur 527 . . . 4 (((𝜓𝜑) ∨ (𝜃𝜒)) ↔ (((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒))))
54bicomi 214 . . 3 ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒))) ↔ ((𝜓𝜑) ∨ (𝜃𝜒)))
6 id 22 . . . . . 6 (𝜃𝜃)
76olci 406 . . . . 5 ((¬ 𝜓𝜑) ∨ (𝜃𝜃))
87biantru 526 . . . 4 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((¬ 𝜓𝜑) ∨ (𝜃𝜃))))
98bicomi 214 . . 3 ((((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((¬ 𝜓𝜑) ∨ (𝜃𝜃))) ↔ ((𝜑𝜓) ∨ (𝜒𝜃)))
105, 9anbi12i 732 . 2 (((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((¬ 𝜓𝜑) ∨ (𝜃𝜃)))) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
111, 10bitri 264 1 ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  if-wif 1011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012
This theorem is referenced by:  ifp1bi  37349
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