Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpbi2 Structured version   Visualization version   GIF version

Theorem ifpbi2 40091
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
Assertion
Ref Expression
ifpbi2 ((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))

Proof of Theorem ifpbi2
StepHypRef Expression
1 imbi2 352 . . 3 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
21anbi1d 632 . 2 ((𝜑𝜓) → (((𝜒𝜑) ∧ (¬ 𝜒𝜃)) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃))))
3 dfifp2 1060 . 2 (if-(𝜒, 𝜑, 𝜃) ↔ ((𝜒𝜑) ∧ (¬ 𝜒𝜃)))
4 dfifp2 1060 . 2 (if-(𝜒, 𝜓, 𝜃) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃)))
52, 3, 43bitr4g 317 1 ((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  if-wif 1058
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 210  df-an 400  df-or 845  df-ifp 1059
This theorem is referenced by:  ifpnot23b  40106
  Copyright terms: Public domain W3C validator