Theorem mpand 433

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 432	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  434  mp2and  437  rspcimedv  2931  ovig  6200  prcdnql  7841  prcunqu  7842  p1le  9169  nnge1  9306  zltp1le  9678  gtndiv  9720  uzss  9922  addlelt  10148  xrre2  10202  xrre3  10203  zltaddlt1le  10389  nn0p1elfzo  10572  zsupcllemstep  10640  modfzo0difsn  10810  seqf1oglem1  10934  leexp2r  11008  expnlbnd2  11081  facavg  11162  wrdred1hash  11326  ccat2s1fvwd  11393  caubnd2  11861  maxleast  11957  mulcn2  12056  cn1lem  12058  climsqz  12079  climsqz2  12080  climcvg1nlem  12093  fsumabs  12210  cvgratnnlemnexp  12269  cvgratnnlemmn  12270  bitsfzolem  12699  bitsfzo  12700  gcdzeq  12777  algcvgblem  12805  algcvga  12807  lcmdvdsb  12840  coprm  12900  pclemub  13044  bldisj  15425  xblm  15441  metss2lem  15521  bdxmet  15525  limccoap  15702  lgsne0  16071  gausslemma2dlem1a  16091  eupth2lemsfi  16633