Theorem mpbi2and 949

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	|- ( ph -> ps )
mpbi2and.2	|- ( ph -> ch )
mpbi2and.3	|- ( ph -> ( ( ps /\ ch ) <-> th ) )

Assertion

Ref	Expression
mpbi2and	|- ( ph -> th )

Proof of Theorem mpbi2and

Step	Hyp	Ref	Expression
1		mpbi2and.1	. . 3 |- ( ph -> ps )
2		mpbi2and.2	. . 3 |- ( ph -> ch )
3	1, 2	jca 306	. 2 |- ( ph -> ( ps /\ ch ) )
4		mpbi2and.3	. 2 |- ( ph -> ( ( ps /\ ch ) <-> th ) )
5	3, 4	mpbid 147	1 |- ( ph -> th )

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  7188  remim  11386  resqrtcl  11555  divalgmod  12453  oddpwdclemxy  12706  divnumden  12733  numdensq  12739  prmdivdiv  12774  4sqlem7  12922  ismgmid2  13428  mnd1  13503  iscmnd  13850  imasring  14042  subrg1  14210  topgele  14718  lmcn2  14969