Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpbi23 Structured version   Visualization version   GIF version

Theorem ifpbi23 41819
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi23 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Proof of Theorem ifpbi23
StepHypRef Expression
1 simpl 484 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
2 simpr 486 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
31, 2ifpbi23d 1081 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  if-wif 1062
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 206  df-an 398  df-or 847  df-ifp 1063
This theorem is referenced by:  ifpnot23d  41831  ifpdfxor  41833  ifpananb  41852  ifpnannanb  41853  ifpxorxorb  41857
  Copyright terms: Public domain W3C validator