Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpimimb Structured version   Visualization version   GIF version

Theorem ifpimimb 36868
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 1008 . 2 (if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ ((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))))
2 imor 427 . . . 4 ((𝜑 → (𝜓𝜒)) ↔ (¬ 𝜑 ∨ (𝜓𝜒)))
3 pm4.8 379 . . . . . 6 ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
43bicomi 213 . . . . 5 𝜑 ↔ (𝜑 → ¬ 𝜑))
54orbi1i 541 . . . 4 ((¬ 𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)))
6 id 22 . . . . . 6 (𝜑𝜑)
76orci 404 . . . . 5 ((𝜑𝜑) ∨ (𝜃𝜒))
87biantru 525 . . . 4 (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ↔ (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))))
92, 5, 83bitri 285 . . 3 ((𝜑 → (𝜓𝜒)) ↔ (((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))))
10 pm4.64 386 . . . 4 ((¬ 𝜑 → (𝜃𝜏)) ↔ (𝜑 ∨ (𝜃𝜏)))
11 pm4.81 380 . . . . . 6 ((¬ 𝜑𝜑) ↔ 𝜑)
1211bicomi 213 . . . . 5 (𝜑 ↔ (¬ 𝜑𝜑))
1312orbi1i 541 . . . 4 ((𝜑 ∨ (𝜃𝜏)) ↔ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))
146orci 404 . . . . 5 ((𝜑𝜑) ∨ (𝜓𝜏))
1514biantrur 526 . . . 4 (((¬ 𝜑𝜑) ∨ (𝜃𝜏)) ↔ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏))))
1610, 13, 153bitri 285 . . 3 ((¬ 𝜑 → (𝜃𝜏)) ↔ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏))))
179, 16anbi12i 729 . 2 (((𝜑 → (𝜓𝜒)) ∧ (¬ 𝜑 → (𝜃𝜏))) ↔ ((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))))
18 ifpim123g 36864 . . 3 ((if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)) ↔ ((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))))
1918bicomi 213 . 2 (((((𝜑 → ¬ 𝜑) ∨ (𝜓𝜒)) ∧ ((𝜑𝜑) ∨ (𝜃𝜒))) ∧ (((𝜑𝜑) ∨ (𝜓𝜏)) ∧ ((¬ 𝜑𝜑) ∨ (𝜃𝜏)))) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
201, 17, 193bitri 285 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 195  wo 382  wa 383  if-wif 1006
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 196  df-or 384  df-an 385  df-ifp 1007
This theorem is referenced by:  ifpororb  36869  ifpbibib  36874
  Copyright terms: Public domain W3C validator