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Theorem ifpbi123d 1079
Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
Hypotheses
Ref Expression
ifpbi123d.1 (𝜑 → (𝜓𝜏))
ifpbi123d.2 (𝜑 → (𝜒𝜂))
ifpbi123d.3 (𝜑 → (𝜃𝜁))
Assertion
Ref Expression
ifpbi123d (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123d
StepHypRef Expression
1 ifpbi123d.1 . . . 4 (𝜑 → (𝜓𝜏))
2 ifpbi123d.2 . . . 4 (𝜑 → (𝜒𝜂))
31, 2imbi12d 348 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜏𝜂)))
4 ifpbi123d.3 . . . 4 (𝜑 → (𝜃𝜁))
51, 4orbi12d 918 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜏𝜁)))
63, 5anbi12d 634 . 2 (𝜑 → (((𝜓𝜒) ∧ (𝜓𝜃)) ↔ ((𝜏𝜂) ∧ (𝜏𝜁))))
7 dfifp3 1065 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∧ (𝜓𝜃)))
8 dfifp3 1065 . 2 (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏𝜂) ∧ (𝜏𝜁)))
96, 7, 83bitr4g 317 1 (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846  if-wif 1062
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 847  df-ifp 1063
This theorem is referenced by:  ifpbi23d  1081  wkslem1  27561  wkslem2  27562  iswlk  27564  wlkres  27624  redwlk  27626  wlkp1lem8  27634  crctcshwlkn0lem4  27763  crctcshwlkn0lem5  27764  crctcshwlkn0lem6  27765  1wlkdlem4  28089  pfxwlk  32668  satfv1fvfmla1  32968  ifpbi123  40691
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