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  2845  ovig  5998  prcdnql  7485  prcunqu  7486  p1le  8808  nnge1  8944  zltp1le  9309  gtndiv  9350  uzss  9550  addlelt  9770  xrre2  9823  xrre3  9824  zltaddlt1le  10009  modfzo0difsn  10397  leexp2r  10576  expnlbnd2  10648  facavg  10728  caubnd2  11128  maxleast  11224  mulcn2  11322  cn1lem  11324  climsqz  11345  climsqz2  11346  climcvg1nlem  11359  fsumabs  11475  cvgratnnlemnexp  11534  cvgratnnlemmn  11535  zsupcllemstep  11948  gcdzeq  12025  algcvgblem  12051  algcvga  12053  lcmdvdsb  12086  coprm  12146  pclemub  12289  bldisj  13940  xblm  13956  metss2lem  14036  bdxmet  14040  limccoap  14186  lgsne0  14478