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Theorem ifpdfbi 1066
Description: Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
Assertion
Ref Expression
ifpdfbi ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Proof of Theorem ifpdfbi
StepHypRef Expression
1 con34b 319 . . 3 ((𝜓𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
21anbi2i 625 . 2 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3 dfbi2 478 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
4 dfifp2 1060 . 2 (if-(𝜑, 𝜓, ¬ 𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
52, 3, 43bitr4i 306 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  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:  wl-df3xor2  34879  wl-2xor  34893  ifpbiidcor  40169  ifpbicor  40170
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