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Theorem ifpid1g 37306
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpid1g ((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒𝜑) ∧ (𝜑𝜓)))

Proof of Theorem ifpid1g
StepHypRef Expression
1 ifpidg 37303 . 2 ((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓)) ∧ ((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒)))))
2 ancom 466 . 2 (((((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓)) ∧ ((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒)))) ↔ (((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒))) ∧ (((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓))))
3 pm4.25 537 . . . . 5 (𝜑 ↔ (𝜑𝜑))
43imbi2i 326 . . . 4 ((𝜒𝜑) ↔ (𝜒 → (𝜑𝜑)))
5 orc 400 . . . . 5 (𝜑 → (𝜑𝜒))
65biantru 526 . . . 4 ((𝜒 → (𝜑𝜑)) ↔ ((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒))))
74, 6bitr2i 265 . . 3 (((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒))) ↔ (𝜒𝜑))
8 pm4.24 674 . . . . 5 (𝜑 ↔ (𝜑𝜑))
98imbi1i 339 . . . 4 ((𝜑𝜓) ↔ ((𝜑𝜑) → 𝜓))
10 simpl 473 . . . . 5 ((𝜑𝜓) → 𝜑)
1110biantrur 527 . . . 4 (((𝜑𝜑) → 𝜓) ↔ (((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓)))
129, 11bitr2i 265 . . 3 ((((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓)) ↔ (𝜑𝜓))
137, 12anbi12i 732 . 2 ((((𝜒 → (𝜑𝜑)) ∧ (𝜑 → (𝜑𝜒))) ∧ (((𝜑𝜓) → 𝜑) ∧ ((𝜑𝜑) → 𝜓))) ↔ ((𝜒𝜑) ∧ (𝜑𝜓)))
141, 2, 133bitri 286 1 ((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒𝜑) ∧ (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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