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Theorem ifpnot23d 36849
 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 36842 . 2 (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
2 notnotb 303 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
3 notnotb 303 . . 3 (𝜒 ↔ ¬ ¬ 𝜒)
4 ifpbi23 36836 . . 3 (((𝜓 ↔ ¬ ¬ 𝜓) ∧ (𝜒 ↔ ¬ ¬ 𝜒)) → (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒)))
52, 3, 4mp2an 704 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ ¬ 𝜓, ¬ ¬ 𝜒))
61, 5bitr4i 266 1 (¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  if-wif 1006 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 196  df-or 384  df-an 385  df-ifp 1007 This theorem is referenced by:  ifpororb  36869
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