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:  fisseneq  6988  exmidapne  7320  prloc  7551  axcaucvglemres  7959  seqf1oglem1  10590  seqf1oglem2  10591  bezoutlemmain  12135  coprm  12282  sqrt2irr  12300  oddprmdvds  12492  lmodfopnelem1  13820  xblss2ps  14572  xblss2  14573  lgsprme0  15158  pw1nct  15493  apdiff  15538