Theorem mpbi2and 956

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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  supisoti  7356  remim  11647  resqrtcl  11817  divalgmod  12719  nnmaxpwlemxy  12973  divnumden  13001  numdensq  13007  prmdivdiv  13044  4sqlem7  13192  ismgmid2  13759  mnd1  13821  resscntz  14166  iscmnd  14191  imasring  14478  subrg1  14648  topgele  15255  lmcn2  15506