Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpbi123 Structured version   Visualization version   GIF version

Theorem ifpbi123 37661
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi123 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Proof of Theorem ifpbi123
StepHypRef Expression
1 simp1 1060 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜑𝜓))
2 simp2 1061 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜒𝜃))
31, 2imbi12d 334 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
41notbid 308 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (¬ 𝜑 ↔ ¬ 𝜓))
5 simp3 1062 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜏𝜂))
64, 5imbi12d 334 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((¬ 𝜑𝜏) ↔ (¬ 𝜓𝜂)))
73, 6anbi12d 747 . 2 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (((𝜑𝜒) ∧ (¬ 𝜑𝜏)) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜂))))
8 dfifp2 1014 . 2 (if-(𝜑, 𝜒, 𝜏) ↔ ((𝜑𝜒) ∧ (¬ 𝜑𝜏)))
9 dfifp2 1014 . 2 (if-(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜂)))
107, 8, 93bitr4g 303 1 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  if-wif 1012  w3a 1037
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 1013  df-3an 1039
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator