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Theorem ifpid2g 37316
 Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpid2g ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))

Proof of Theorem ifpid2g
StepHypRef Expression
1 ifpidg 37314 . 2 ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓)) ∧ ((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒)))))
2 simpr 477 . . . 4 ((𝜑𝜓) → 𝜓)
32, 2pm3.2i 471 . . 3 (((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓))
43biantrur 527 . 2 (((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒))) ↔ ((((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓)) ∧ ((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒)))))
5 ancom 466 . 2 (((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒))) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
61, 4, 53bitr2i 288 1 ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
 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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