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Theorem ifpnot23 38140
Description: Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpnot23 (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Proof of Theorem ifpnot23
StepHypRef Expression
1 ianor 508 . . . 4 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2 pm4.55 514 . . . 4 (¬ (¬ 𝜑𝜒) ↔ (𝜑 ∨ ¬ 𝜒))
31, 2anbi12i 733 . . 3 ((¬ (𝜑𝜓) ∧ ¬ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
4 ioran 510 . . 3 (¬ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ (¬ (𝜑𝜓) ∧ ¬ (¬ 𝜑𝜒)))
5 dfifp4 1036 . . 3 (if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (𝜑 ∨ ¬ 𝜒)))
63, 4, 53bitr4i 292 . 2 (¬ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
7 df-ifp 1033 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
86, 7xchnxbir 322 1 (¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383  if-wif 1032
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 384  df-an 385  df-ifp 1033
This theorem is referenced by:  ifpnotnotb  38141  ifpnorcor  38142  ifpnancor  38143  ifpnot23b  38144  ifpnot23c  38146  ifpnot23d  38147  ifpdfnan  38148  ifpdfxor  38149  ifpor123g  38170
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