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Theorem ifpbi2 39710
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 350 . . 3 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
21anbi1d 629 . 2 ((𝜑𝜓) → (((𝜒𝜑) ∧ (¬ 𝜒𝜃)) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃))))
3 dfifp2 1056 . 2 (if-(𝜒, 𝜑, 𝜃) ↔ ((𝜒𝜑) ∧ (¬ 𝜒𝜃)))
4 dfifp2 1056 . 2 (if-(𝜒, 𝜓, 𝜃) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃)))
52, 3, 43bitr4g 315 1 ((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  if-wif 1054
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 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by:  ifpnot23b  39726
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