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Theorem 3biant1d 1433
 Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 527. (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 528 . 2 (𝜑 → ((𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓))))
3 3anass 1035 . 2 ((𝜃𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓)))
42, 3syl6bbr 277 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031 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 196  df-an 385  df-3an 1033 This theorem is referenced by:  metuel2  22180  itgsubst  23616  clwlkisclwwlk  26317  itg2addnclem2  32632  clwlkclwwlk  41211
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