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Theorem dfbi 664
Description: Definition df-bi 197 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)
Assertion
Ref Expression
dfbi (((𝜑𝜓) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓)))

Proof of Theorem dfbi
StepHypRef Expression
1 dfbi2 663 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21biimpi 206 . 2 ((𝜑𝜓) → ((𝜑𝜓) ∧ (𝜓𝜑)))
31biimpri 218 . 2 (((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓))
42, 3pm3.2i 470 1 (((𝜑𝜓) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383
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-an 385
This theorem is referenced by: (None)
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