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Theorem ifpxorcor 40182
 Description: Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpxorcor (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))

Proof of Theorem ifpxorcor
StepHypRef Expression
1 ifpbicor 40181 . 2 (if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
2 notnotb 318 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
3 ifpbi3 40174 . . 3 ((𝜓 ↔ ¬ ¬ 𝜓) → (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓)))
42, 3ax-mp 5 . 2 (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓))
5 ifpn 1069 . 2 (if-(𝜓, ¬ 𝜑, 𝜑) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
61, 4, 53bitr4i 306 1 (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  if-wif 1058 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 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by: (None)
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