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Theorem ifpbi3 38310
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
Assertion
Ref Expression
ifpbi3 ((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

Proof of Theorem ifpbi3
StepHypRef Expression
1 imbi2 339 . . 3 ((𝜑𝜓) → ((¬ 𝜒𝜑) ↔ (¬ 𝜒𝜓)))
21anbi2d 616 . 2 ((𝜑𝜓) → (((𝜒𝜃) ∧ (¬ 𝜒𝜑)) ↔ ((𝜒𝜃) ∧ (¬ 𝜒𝜓))))
3 dfifp2 1080 . 2 (if-(𝜒, 𝜃, 𝜑) ↔ ((𝜒𝜃) ∧ (¬ 𝜒𝜑)))
4 dfifp2 1080 . 2 (if-(𝜒, 𝜃, 𝜓) ↔ ((𝜒𝜃) ∧ (¬ 𝜒𝜓)))
52, 3, 43bitr4g 305 1 ((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  if-wif 1078
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 198  df-an 385  df-or 866  df-ifp 1079
This theorem is referenced by:  ifpxorcor  38319  ifpnot23c  38327  ifpdfnan  38329
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