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Theorem ifpid3g 37345
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpid3g ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)))

Proof of Theorem ifpid3g
StepHypRef Expression
1 olc 399 . . 3 (𝜒 → (𝜑𝜒))
21, 1pm3.2i 471 . 2 ((𝜒 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜒)))
3 ifpidg 37344 . 2 ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)) ∧ ((𝜒 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜒)))))
42, 3mpbiran2 953 1 ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  if-wif 1011
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 197  df-or 385  df-an 386  df-ifp 1012
This theorem is referenced by: (None)
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