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Theorem mpbirand 531
 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 530 . 2 (𝜑 → (𝜃 ↔ (𝜒𝜃)))
41, 3bitr4d 271 1 (𝜑 → (𝜓𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383 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 385 This theorem is referenced by:  3anibar  1390  rmob2  3660  opbrop  5343  opelresi  5554  iscvs  23098  isspthonpth  26826  clwlkclwwlkflem  27098  clwwlkwwlksb  27155  esum2dlem  30434  ntrclselnel1  38826  ntrneicls00  38858  vonvolmbl  41350
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