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Theorem ifpbi23 43376
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 482 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
2 simpr 484 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
31, 2ifpbi23d 1080 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  if-wif 1063
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 207  df-an 396  df-or 847  df-ifp 1064
This theorem is referenced by:  ifpnot23d  43388  ifpdfxor  43390  ifpananb  43409  ifpnannanb  43410  ifpxorxorb  43414
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