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Theorem ifpbi1 40101
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
Assertion
Ref Expression
ifpbi1 ((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))

Proof of Theorem ifpbi1
StepHypRef Expression
1 imbi1 351 . . 3 ((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
2 notbi 322 . . . . 5 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
32biimpi 219 . . . 4 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43imbi1d 345 . . 3 ((𝜑𝜓) → ((¬ 𝜑𝜃) ↔ (¬ 𝜓𝜃)))
51, 4anbi12d 633 . 2 ((𝜑𝜓) → (((𝜑𝜒) ∧ (¬ 𝜑𝜃)) ↔ ((𝜓𝜒) ∧ (¬ 𝜓𝜃))))
6 dfifp2 1060 . 2 (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑𝜒) ∧ (¬ 𝜑𝜃)))
7 dfifp2 1060 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∧ (¬ 𝜓𝜃)))
85, 6, 73bitr4g 317 1 ((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 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:  ifpimim  40133  axfrege52a  40474
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