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Theorem bj-dfbi6 32897
Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
Assertion
Ref Expression
bj-dfbi6 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Proof of Theorem bj-dfbi6
StepHypRef Expression
1 bj-dfbi5 32896 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))
2 id 22 . . . 4 (((𝜑𝜓) → (𝜑𝜓)) → ((𝜑𝜓) → (𝜑𝜓)))
3 animorr 959 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
42, 3impbid1 215 . . 3 (((𝜑𝜓) → (𝜑𝜓)) → ((𝜑𝜓) ↔ (𝜑𝜓)))
5 biimp 205 . . 3 (((𝜑𝜓) ↔ (𝜑𝜓)) → ((𝜑𝜓) → (𝜑𝜓)))
64, 5impbii 199 . 2 (((𝜑𝜓) → (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
71, 6bitri 264 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 834
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 383  df-or 835
This theorem is referenced by: (None)
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