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Theorem bj-dfbi4 31573
Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
Assertion
Ref Expression
bj-dfbi4 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Proof of Theorem bj-dfbi4
StepHypRef Expression
1 dfbi3 932 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 pm4.56 514 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
32orbi2i 539 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
41, 3bitri 262 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wo 381  wa 382
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 195  df-or 383  df-an 384
This theorem is referenced by:  bj-dfbi5  31574
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