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Theorem ifpbi2 36830
 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 337 . . 3 ((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
21anbi1d 737 . 2 ((𝜑𝜓) → (((𝜒𝜑) ∧ (¬ 𝜒𝜃)) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃))))
3 dfifp2 1008 . 2 (if-(𝜒, 𝜑, 𝜃) ↔ ((𝜒𝜑) ∧ (¬ 𝜒𝜃)))
4 dfifp2 1008 . 2 (if-(𝜒, 𝜓, 𝜃) ↔ ((𝜒𝜓) ∧ (¬ 𝜒𝜃)))
52, 3, 43bitr4g 302 1 ((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006 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 196  df-or 384  df-an 385  df-ifp 1007 This theorem is referenced by:  ifpnot23b  36846
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