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Theorem ifpn 1021
Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
ifpn (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Proof of Theorem ifpn
StepHypRef Expression
1 notnotb 304 . . . 4 (𝜑 ↔ ¬ ¬ 𝜑)
21imbi1i 339 . . 3 ((𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓))
32anbi2ci 731 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜒) ∧ (¬ ¬ 𝜑𝜓)))
4 dfifp2 1013 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
5 dfifp2 1013 . 2 (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑𝜒) ∧ (¬ ¬ 𝜑𝜓)))
63, 4, 53bitr4i 292 1 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  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:  ifpfal  1023  ifpdfbi  37334  ifpxorcor  37337
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