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Theorem dfifp3 1057
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
dfifp3 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem dfifp3
StepHypRef Expression
1 dfifp2 1056 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 pm4.64 843 . . 3 ((¬ 𝜑𝜒) ↔ (𝜑𝜒))
32anbi2i 622 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
41, 3bitri 276 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  if-wif 1054
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 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by:  dfifp4  1058  ifptru  1065
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