Theorem syl5com 29

Description: Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)

Hypotheses

Ref	Expression
syl5com.1	|- ( ph -> ps )
syl5com.2	|- ( ch -> ( ps -> th ) )

Assertion

Ref	Expression
syl5com	|- ( ph -> ( ch -> th ) )

Proof of Theorem syl5com

Step	Hyp	Ref	Expression
1		syl5com.1	. . 3 |- ( ph -> ps )
2	1	a1d 22	. 2 |- ( ph -> ( ch -> ps ) )
3		syl5com.2	. 2 |- ( ch -> ( ps -> th ) )
4	2, 3	sylcom 28	1 |- ( ph -> ( ch -> 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:  com12  30  syl5  32  pm2.6dc  847  pm5.11dc  894  ax16i  1830  mor  2041  ceqsalg  2714  cgsexg  2721  cgsex2g  2722  cgsex4g  2723  spc2egv  2775  spc2gv  2776  spc3egv  2777  spc3gv  2778  disjne  3416  uneqdifeqim  3448  eqifdc  3506  triun  4039  sucssel  4346  ordsucg  4418  regexmidlem1  4448  relresfld  5068  relcoi1  5070  fornex  6013  f1dmex  6014  dom2d  6667  findcard  6782  nneo  9161  zeo2  9164  uznfz  9890  difelfzle  9918  ssfzo12  10008  facndiv  10492  fisumcom2  11214  ndvdssub  11634  bezoutlembi  11700  eucalglt  11745  prmind2  11808  coprm  11829  inopn  12180  basis1  12224  tgss  12242  tgcl  12243  xmeteq0  12538  blssexps  12608  blssex  12609  mopni3  12663  neibl  12670  metss  12673  metcnp3  12690  bj-indsuc  13136  bj-nntrans  13159