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  2923  ovig  6175  prcdnql  7799  prcunqu  7800  p1le  9123  nnge1  9260  zltp1le  9632  gtndiv  9673  uzss  9875  addlelt  10101  xrre2  10154  xrre3  10155  zltaddlt1le  10341  nn0p1elfzo  10521  zsupcllemstep  10589  modfzo0difsn  10757  seqf1oglem1  10881  leexp2r  10955  expnlbnd2  11027  facavg  11108  wrdred1hash  11268  ccat2s1fvwd  11335  caubnd2  11802  maxleast  11898  mulcn2  11997  cn1lem  11999  climsqz  12020  climsqz2  12021  climcvg1nlem  12034  fsumabs  12151  cvgratnnlemnexp  12210  cvgratnnlemmn  12211  bitsfzolem  12640  bitsfzo  12641  gcdzeq  12718  algcvgblem  12746  algcvga  12748  lcmdvdsb  12781  coprm  12841  pclemub  12985  bldisj  15266  xblm  15282  metss2lem  15362  bdxmet  15366  limccoap  15543  lgsne0  15911  gausslemma2dlem1a  15931  eupth2lemsfi  16473