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Theorem ifpan23 39686
Description: Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpan23 ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))

Proof of Theorem ifpan23
StepHypRef Expression
1 ifpan123g 39685 . 2 ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ (((¬ 𝜑𝜓) ∧ (𝜑𝜒)) ∧ ((¬ 𝜑𝜃) ∧ (𝜑𝜏))))
2 an4 652 . 2 ((((¬ 𝜑𝜓) ∧ (𝜑𝜒)) ∧ ((¬ 𝜑𝜃) ∧ (𝜑𝜏))) ↔ (((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜃)) ∧ ((𝜑𝜒) ∧ (𝜑𝜏))))
3 dfifp4 1060 . . 3 (if-(𝜑, (𝜓𝜃), (𝜒𝜏)) ↔ ((¬ 𝜑 ∨ (𝜓𝜃)) ∧ (𝜑 ∨ (𝜒𝜏))))
4 ordi 1001 . . . 4 ((¬ 𝜑 ∨ (𝜓𝜃)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜃)))
5 ordi 1001 . . . 4 ((𝜑 ∨ (𝜒𝜏)) ↔ ((𝜑𝜒) ∧ (𝜑𝜏)))
64, 5anbi12i 626 . . 3 (((¬ 𝜑 ∨ (𝜓𝜃)) ∧ (𝜑 ∨ (𝜒𝜏))) ↔ (((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜃)) ∧ ((𝜑𝜒) ∧ (𝜑𝜏))))
73, 6bitr2i 277 . 2 ((((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜃)) ∧ ((𝜑𝜒) ∧ (𝜑𝜏))) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))
81, 2, 73bitri 298 1 ((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wo 843  if-wif 1056
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 208  df-an 397  df-or 844  df-ifp 1057
This theorem is referenced by:  ifpdfbi  39700  ifpdfxor  39714
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