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Theorem ifpor123g 40132
Description: Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpor123g ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))

Proof of Theorem ifpor123g
StepHypRef Expression
1 df-or 845 . . . 4 ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ (¬ if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)))
2 ifpnot23 40102 . . . . 5 (¬ if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜑, ¬ 𝜒, ¬ 𝜏))
32imbi1i 353 . . . 4 ((¬ if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ (if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)))
41, 3bitri 278 . . 3 ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ (if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)))
5 ifpim123g 40124 . . 3 ((if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (¬ 𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (¬ 𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (¬ 𝜏𝜂)))))
64, 5bitri 278 . 2 ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (¬ 𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (¬ 𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (¬ 𝜏𝜂)))))
7 pm4.64 846 . . . . 5 ((¬ 𝜒𝜃) ↔ (𝜒𝜃))
87orbi2i 910 . . . 4 (((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒𝜃)) ↔ ((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)))
9 pm4.64 846 . . . . 5 ((¬ 𝜏𝜃) ↔ (𝜏𝜃))
109orbi2i 910 . . . 4 (((𝜓𝜑) ∨ (¬ 𝜏𝜃)) ↔ ((𝜓𝜑) ∨ (𝜏𝜃)))
118, 10anbi12i 629 . . 3 ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (¬ 𝜏𝜃))) ↔ (((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))))
12 pm4.64 846 . . . . 5 ((¬ 𝜒𝜂) ↔ (𝜒𝜂))
1312orbi2i 910 . . . 4 (((𝜑𝜓) ∨ (¬ 𝜒𝜂)) ↔ ((𝜑𝜓) ∨ (𝜒𝜂)))
14 pm4.64 846 . . . . 5 ((¬ 𝜏𝜂) ↔ (𝜏𝜂))
1514orbi2i 910 . . . 4 (((¬ 𝜓𝜑) ∨ (¬ 𝜏𝜂)) ↔ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))
1613, 15anbi12i 629 . . 3 ((((𝜑𝜓) ∨ (¬ 𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (¬ 𝜏𝜂))) ↔ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂))))
1711, 16anbi12i 629 . 2 (((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (¬ 𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (¬ 𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (¬ 𝜏𝜂)))) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))
186, 17bitri 278 1 ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058
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 210  df-an 400  df-or 845  df-ifp 1059
This theorem is referenced by: (None)
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