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Theorem mpbi2and 861
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
Hypotheses
Ref Expression
mpbi2and.1 (𝜑𝜓)
mpbi2and.2 (𝜑𝜒)
mpbi2and.3 (𝜑 → ((𝜓𝜒) ↔ 𝜃))
Assertion
Ref Expression
mpbi2and (𝜑𝜃)

Proof of Theorem mpbi2and
StepHypRef Expression
1 mpbi2and.1 . . 3 (𝜑𝜓)
2 mpbi2and.2 . . 3 (𝜑𝜒)
31, 2jca 294 . 2 (𝜑 → (𝜓𝜒))
4 mpbi2and.3 . 2 (𝜑 → ((𝜓𝜒) ↔ 𝜃))
53, 4mpbid 139 1 (𝜑𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  supisoti  6413  remim  9681  resqrtcl  9848  oddpwdclemxy  10229
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