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Theorem mpbirand 530
Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
mpbirand.1 (𝜑𝜒)
mpbirand.2 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
Assertion
Ref Expression
mpbirand (𝜑 → (𝜓𝜃))

Proof of Theorem mpbirand
StepHypRef Expression
1 mpbirand.2 . 2 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
2 mpbirand.1 . . 3 (𝜑𝜒)
32biantrurd 529 . 2 (𝜑 → (𝜃 ↔ (𝜒𝜃)))
41, 3bitr4d 271 1 (𝜑 → (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384
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 386
This theorem is referenced by:  3anibar  1227  rmob2  3512  opbrop  5159  opelresi  5367  iscvs  22835  isspthonpth  26514  esum2dlem  29932  ntrclselnel1  37834  ntrneicls00  37866  vonvolmbl  40179
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