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Theorem 3biant1d 1471
Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 532. (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 533 . 2 (𝜑 → ((𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓))))
3 3anass 1089 . 2 ((𝜃𝜒𝜓) ↔ (𝜃 ∧ (𝜒𝜓)))
42, 3syl6bbr 290 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081
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 208  df-an 397  df-3an 1083
This theorem is referenced by:  metuel2  23090  itgsubst  24561  clwlkclwwlk  27694  dfgcd3  34474  itg2addnclem2  34811
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