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Theorem ifpbibib 37375
Description: Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpbibib (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))

Proof of Theorem ifpbibib
StepHypRef Expression
1 dfifp2 1013 . 2 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ ((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))))
2 dfbi2 659 . . . . . 6 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
32imbi2i 326 . . . . 5 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) ∧ (𝜒𝜓))))
4 jcab 906 . . . . 5 ((𝜑 → ((𝜓𝜒) ∧ (𝜒𝜓))) ↔ ((𝜑 → (𝜓𝜒)) ∧ (𝜑 → (𝜒𝜓))))
53, 4bitri 264 . . . 4 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑 → (𝜓𝜒)) ∧ (𝜑 → (𝜒𝜓))))
6 dfbi2 659 . . . . . 6 ((𝜃𝜏) ↔ ((𝜃𝜏) ∧ (𝜏𝜃)))
76imbi2i 326 . . . . 5 ((¬ 𝜑 → (𝜃𝜏)) ↔ (¬ 𝜑 → ((𝜃𝜏) ∧ (𝜏𝜃))))
8 jcab 906 . . . . 5 ((¬ 𝜑 → ((𝜃𝜏) ∧ (𝜏𝜃))) ↔ ((¬ 𝜑 → (𝜃𝜏)) ∧ (¬ 𝜑 → (𝜏𝜃))))
97, 8bitri 264 . . . 4 ((¬ 𝜑 → (𝜃𝜏)) ↔ ((¬ 𝜑 → (𝜃𝜏)) ∧ (¬ 𝜑 → (𝜏𝜃))))
105, 9anbi12i 732 . . 3 (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ↔ (((𝜑 → (𝜓𝜒)) ∧ (𝜑 → (𝜒𝜓))) ∧ ((¬ 𝜑 → (𝜃𝜏)) ∧ (¬ 𝜑 → (𝜏𝜃)))))
11 an4 864 . . 3 ((((𝜑 → (𝜓𝜒)) ∧ (𝜑 → (𝜒𝜓))) ∧ ((¬ 𝜑 → (𝜃𝜏)) ∧ (¬ 𝜑 → (𝜏𝜃)))) ↔ (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ∧ ((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃)))))
1210, 11bitri 264 . 2 (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ↔ (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ∧ ((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃)))))
13 dfifp2 1013 . . . . 5 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ ((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))))
14 ifpimimb 37369 . . . . 5 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
1513, 14bitr3i 266 . . . 4 (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
16 dfifp2 1013 . . . . 5 (if-(𝜑, (𝜒𝜓), (𝜏𝜃)) ↔ ((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃))))
17 ifpimimb 37369 . . . . 5 (if-(𝜑, (𝜒𝜓), (𝜏𝜃)) ↔ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃)))
1816, 17bitr3i 266 . . . 4 (((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃))) ↔ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃)))
1915, 18anbi12i 732 . . 3 ((((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ∧ ((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃)))) ↔ ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ∧ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃))))
20 dfbi2 659 . . 3 ((if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)) ↔ ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ∧ (if-(𝜑, 𝜒, 𝜏) → if-(𝜑, 𝜓, 𝜃))))
2119, 20bitr4i 267 . 2 ((((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ∧ ((𝜑 → (𝜒𝜓)) ∧ (¬ 𝜑 → (𝜏𝜃)))) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))
221, 12, 213bitri 286 1 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  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:  ifpxorxorb  37376
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