Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpbi2and GIF version

Theorem mpbi2and 928
 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 304 . 2 (𝜑 → (𝜓𝜒))
4 mpbi2and.3 . 2 (𝜑 → ((𝜓𝜒) ↔ 𝜃))
53, 4mpbid 146 1 (𝜑𝜃)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  supisoti  6904  remim  10663  resqrtcl  10832  divalgmod  11658  oddpwdclemxy  11881  divnumden  11908  numdensq  11914  topgele  12233  lmcn2  12486
 Copyright terms: Public domain W3C validator