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Theorem dfifp5 1048
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp5 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Proof of Theorem dfifp5
StepHypRef Expression
1 dfifp2 1045 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 imor 839 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32anbi1i 614 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
41, 3bitri 267 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833  if-wif 1043
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 199  df-an 388  df-or 834  df-ifp 1044
This theorem is referenced by: (None)
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