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Theorem ifpxorcor 38361
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 38360 . 2 (if-(𝜑, ¬ 𝜓, ¬ ¬ 𝜓) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
2 notnotb 305 . . 3 (𝜓 ↔ ¬ ¬ 𝜓)
3 ifpbi3 38352 . . 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 1086 . 2 (if-(𝜓, ¬ 𝜑, 𝜑) ↔ if-(¬ 𝜓, 𝜑, ¬ 𝜑))
61, 4, 53bitr4i 293 1 (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  if-wif 1076
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 198  df-an 384  df-or 864  df-ifp 1077  df-tru 1637
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator