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Theorem anifp 1067
Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1068. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
anifp ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))

Proof of Theorem anifp
StepHypRef Expression
1 olc 865 . . 3 (𝜓 → (¬ 𝜑𝜓))
2 olc 865 . . 3 (𝜒 → (𝜑𝜒))
31, 2anim12i 615 . 2 ((𝜓𝜒) → ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
4 dfifp4 1062 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
53, 4sylibr 237 1 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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:  bj-consensus  33996  bj-consensusALT  33997  axfrege58a  40520
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