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  7200  remim  11411  resqrtcl  11580  divalgmod  12478  oddpwdclemxy  12731  divnumden  12758  numdensq  12764  prmdivdiv  12799  4sqlem7  12947  ismgmid2  13453  mnd1  13528  iscmnd  13875  imasring  14067  subrg1  14235  topgele  14743  lmcn2  14994