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Theorem 3biant1d 1479
Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 535. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
Hypothesis
Ref Expression
3biantd.1 (𝜑𝜃)
Assertion
Ref Expression
3biant1d (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))

Proof of Theorem 3biant1d
StepHypRef Expression
1 3biantd.1 . . 3 (𝜑𝜃)
21biantrurd 536 . 2 (𝜑 → ((𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓))))
3 3anass 1096 . 2 ((𝜃𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓)))
42, 3bitr4di 292 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088
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-3an 1090
This theorem is referenced by:  metuel2  23321  itgsubst  24804  clwlkclwwlk  27942  dfgcd3  35138  itg2addnclem2  35475
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