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Theorem dfifp4 1015
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 1014 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
2 imor 428 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32anbi1i 730 . 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 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:  anifp  1019  ifpan123g  37622  ifpan23  37623  ifpdfor2  37624  ifpdfor  37628  ifpim1  37632  ifpnot  37633  ifpid2  37634  ifpim2  37635  ifpnot23  37642  ifpidg  37655  ifpim123g  37664  ifpimim  37673
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