Theorem mpbi2and 945

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  7111  remim  11113  resqrtcl  11282  divalgmod  12180  oddpwdclemxy  12433  divnumden  12460  numdensq  12466  prmdivdiv  12501  4sqlem7  12649  ismgmid2  13154  mnd1  13229  iscmnd  13576  imasring  13768  subrg1  13935  topgele  14443  lmcn2  14694