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Theorem ifpimimb 37330
Description: Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
Assertion
Ref Expression
ifpimimb (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))

Proof of Theorem ifpimimb
StepHypRef Expression
1 dfifp2 1013 . 2 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ ((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))))
2 imor 428 . . . 4 ((𝜑 → (𝜓𝜒)) ↔ (¬ 𝜑 ∨ (𝜓𝜒)))
3 pm4.8 380 . . . . . 6 ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
43bicomi 214 . . . . 5 𝜑 ↔ (𝜑 → ¬ 𝜑))
54orbi1i 542 . . . 4 ((¬ 𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)))
6 id 22 . . . . . 6 (𝜑𝜑)
76orci 405 . . . . 5 ((𝜑𝜑) ∨ (𝜃𝜒))
87biantru 526 . . . 4 (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ↔ (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))))
92, 5, 83bitri 286 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))))
10 pm4.64 387 . . . 4 ((¬ 𝜑 → (𝜃𝜏)) ↔ (𝜑 ∨ (𝜃𝜏)))
11 pm4.81 381 . . . . . 6 ((¬ 𝜑𝜑) ↔ 𝜑)
1211bicomi 214 . . . . 5 (𝜑 ↔ (¬ 𝜑𝜑))
1312orbi1i 542 . . . 4 ((𝜑 ∨ (𝜃𝜏)) ↔ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))
146orci 405 . . . . 5 ((𝜑𝜑) ∨ (𝜓𝜏))
1514biantrur 527 . . . 4 (((¬ 𝜑𝜑) ∨ (𝜃𝜏)) ↔ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏))))
1610, 13, 153bitri 286 . . 3 ((¬ 𝜑 → (𝜃𝜏)) ↔ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏))))
179, 16anbi12i 732 . 2 (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ↔ ((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))))
18 ifpim123g 37326 . . 3 ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ↔ ((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))))
1918bicomi 214 . 2 (((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
201, 17, 193bitri 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  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:  ifpororb  37331  ifpbibib  37336
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