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Theorem bj-animbi 34158
 Description: Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
Assertion
Ref Expression
bj-animbi ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))

Proof of Theorem bj-animbi
StepHypRef Expression
1 simpl 486 . . 3 ((𝜑𝜓) → 𝜑)
2 pm3.4 809 . . 3 ((𝜑𝜓) → (𝜑𝜓))
31, 22thd 268 . 2 ((𝜑𝜓) → (𝜑 ↔ (𝜑𝜓)))
4 biimp 218 . . . . 5 ((𝜑 ↔ (𝜑𝜓)) → (𝜑 → (𝜑𝜓)))
54pm2.43d 53 . . . 4 ((𝜑 ↔ (𝜑𝜓)) → (𝜑𝜓))
6 biimpr 223 . . . 4 ((𝜑 ↔ (𝜑𝜓)) → ((𝜑𝜓) → 𝜑))
75, 6mpd 15 . . 3 ((𝜑 ↔ (𝜑𝜓)) → 𝜑)
87, 5jcai 520 . 2 ((𝜑 ↔ (𝜑𝜓)) → (𝜑𝜓))
93, 8impbii 212 1 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399 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 This theorem is referenced by:  bj-currypara  34159
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