Theorem mpand 429

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpand.1	|- ( ph -> ps )
mpand.2	|- ( ph -> ( ( ps /\ ch ) -> th ) )

Assertion

Ref	Expression
mpand	|- ( ph -> ( ch -> th ) )

Proof of Theorem mpand

Step	Hyp	Ref	Expression
1		mpand.1	. 2 |- ( ph -> ps )
2		mpand.2	. . 3 |- ( ph -> ( ( ps /\ ch ) -> th ) )
3	2	ancomsd 269	. 2 |- ( ph -> ( ( ch /\ ps ) -> th ) )
4	1, 3	mpan2d 428	1 |- ( ph -> ( ch -> th ) )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mpani  430  mp2and  433  rspcimedv  2909  ovig  6125  prcdnql  7667  prcunqu  7668  p1le  8992  nnge1  9129  zltp1le  9497  gtndiv  9538  uzss  9739  addlelt  9960  xrre2  10013  xrre3  10014  zltaddlt1le  10199  zsupcllemstep  10444  modfzo0difsn  10612  seqf1oglem1  10736  leexp2r  10810  expnlbnd2  10882  facavg  10963  wrdred1hash  11110  caubnd2  11623  maxleast  11719  mulcn2  11818  cn1lem  11820  climsqz  11841  climsqz2  11842  climcvg1nlem  11855  fsumabs  11971  cvgratnnlemnexp  12030  cvgratnnlemmn  12031  bitsfzolem  12460  bitsfzo  12461  gcdzeq  12538  algcvgblem  12566  algcvga  12568  lcmdvdsb  12601  coprm  12661  pclemub  12805  bldisj  15069  xblm  15085  metss2lem  15165  bdxmet  15169  limccoap  15346  lgsne0  15711  gausslemma2dlem1a  15731