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Theorem dfifp4 1054
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
dfifp4 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem dfifp4
StepHypRef Expression
1 dfifp3 1053 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
2 imor 427 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32anbi1i 733 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
41, 3bitri 264 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  if-wif 1050
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 384  df-an 385  df-ifp 1051
This theorem is referenced by:  anifp  1058  ifpan123g  38274  ifpan23  38275  ifpdfor2  38276  ifpdfor  38280  ifpim1  38284  ifpnot  38285  ifpid2  38286  ifpim2  38287  ifpnot23  38294  ifpidg  38307  ifpim123g  38316  ifpimim  38325
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