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Theorem ifpnot23d 38353
Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpnot23d (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))

Proof of Theorem ifpnot23d
StepHypRef Expression
1 ifpnot23 38346 . 2 (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
2 notnotb 304 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
3 notnotb 304 . . 3 (𝜒 ↔ ¬ ¬ 𝜒)
4 ifpbi23 38340 . . 3 (((𝜓 ↔ ¬ ¬ 𝜓) ∧ (𝜒 ↔ ¬ ¬ 𝜒)) → (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒)))
52, 3, 4mp2an 672 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
61, 5bitr4i 267 1 (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  if-wif 1049
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-an 383  df-or 837  df-ifp 1050
This theorem is referenced by:  ifpororb  38373
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