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Theorem mpbi2and 943
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 306 . 2 (𝜑 → (𝜓𝜒))
4 mpbi2and.3 . 2 (𝜑 → ((𝜓𝜒) ↔ 𝜃))
53, 4mpbid 147 1 (𝜑𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  supisoti  7002  remim  10840  resqrtcl  11009  divalgmod  11902  oddpwdclemxy  12139  divnumden  12166  numdensq  12172  prmdivdiv  12207  4sqlem7  12352  ismgmid2  12678  mnd1  12724  iscmnd  12915  topgele  13160  lmcn2  13413
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