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Theorem dfifp3 1061
 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 1060 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 pm4.64 846 . . 3 ((¬ 𝜑𝜒) ↔ (𝜑𝜒))
32anbi2i 625 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
41, 3bitri 278 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058 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 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by:  dfifp4  1062  ifpn  1069  ifptru  1071  ifpbi123d  1075  wl-2mintru1  34926
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