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  2910  ovig  6138  prcdnql  7694  prcunqu  7695  p1le  9019  nnge1  9156  zltp1le  9524  gtndiv  9565  uzss  9767  addlelt  9993  xrre2  10046  xrre3  10047  zltaddlt1le  10232  zsupcllemstep  10479  modfzo0difsn  10647  seqf1oglem1  10771  leexp2r  10845  expnlbnd2  10917  facavg  10998  wrdred1hash  11147  ccat2s1fvwd  11214  caubnd2  11668  maxleast  11764  mulcn2  11863  cn1lem  11865  climsqz  11886  climsqz2  11887  climcvg1nlem  11900  fsumabs  12016  cvgratnnlemnexp  12075  cvgratnnlemmn  12076  bitsfzolem  12505  bitsfzo  12506  gcdzeq  12583  algcvgblem  12611  algcvga  12613  lcmdvdsb  12646  coprm  12706  pclemub  12850  bldisj  15115  xblm  15131  metss2lem  15211  bdxmet  15215  limccoap  15392  lgsne0  15757  gausslemma2dlem1a  15777