Theorem mp2d 47

Description: A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Hypotheses

Ref	Expression
mp2d.1	|- ( ph -> ps )
mp2d.2	|- ( ph -> ch )
mp2d.3	|- ( ph -> ( ps -> ( ch -> th ) ) )

Assertion

Ref	Expression
mp2d	|- ( ph -> th )

Proof of Theorem mp2d

Step	Hyp	Ref	Expression
1		mp2d.1	. 2 |- ( ph -> ps )
2		mp2d.2	. . 3 |- ( ph -> ch )
3		mp2d.3	. . 3 |- ( ph -> ( ps -> ( ch -> th ) ) )
4	2, 3	mpid 42	. 2 |- ( ph -> ( ps -> th ) )
5	1, 4	mpd 13	1 |- ( ph -> th )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  riotaeqimp  5996  fisseneq  7127  exmidapne  7479  prloc  7711  axcaucvglemres  8119  seqf1oglem1  10782  seqf1oglem2  10783  wrdind  11307  wrd2ind  11308  bezoutlemmain  12587  coprm  12734  sqrt2irr  12752  oddprmdvds  12945  lmodfopnelem1  14357  xblss2ps  15147  xblss2  15148  perfectlem2  15743  lgsprme0  15790  pw1nct  16655  apdiff  16703