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Theorem ifpid2g 40196
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 40194 . 2 ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓)) ∧ ((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒)))))
2 simpr 488 . . . 4 ((𝜑𝜓) → 𝜓)
32, 2pm3.2i 474 . . 3 (((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓))
43biantrur 534 . 2 (((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒))) ↔ ((((𝜑𝜓) → 𝜓) ∧ ((𝜑𝜓) → 𝜓)) ∧ ((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒)))))
5 ancom 464 . 2 (((𝜒 → (𝜑𝜓)) ∧ (𝜓 → (𝜑𝜒))) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
61, 4, 53bitr2i 302 1 ((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  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: (None)
  Copyright terms: Public domain W3C validator