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Theorem dfifp3 1035
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 1034 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 pm4.64 386 . . 3 ((¬ 𝜑𝜒) ↔ (𝜑𝜒))
32anbi2i 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 382  wa 383  if-wif 1032
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 1033
This theorem is referenced by:  dfifp4  1036  ifptru  1043
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