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Theorem ifpnot23c 38348
Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpnot23c (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))

Proof of Theorem ifpnot23c
StepHypRef Expression
1 ifpnot23 38342 . 2 (¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒))
2 notnotb 304 . . 3 (𝜒 ↔ ¬ ¬ 𝜒)
3 ifpbi3 38331 . . 3 ((𝜒 ↔ ¬ ¬ 𝜒) → (if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒)))
42, 3ax-mp 5 . 2 (if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜒))
51, 4bitr4i 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:  ifpnim1  38361  ifpnim2  38363
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