Theorem sylcom 28

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)

Hypotheses

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

Assertion

Ref	Expression
sylcom	|- ( ph -> ( ps -> th ) )

Proof of Theorem sylcom

Step	Hyp	Ref	Expression
1		sylcom.1	. 2 |- ( ph -> ( ps -> ch ) )
2		sylcom.2	. . 3 |- ( ps -> ( ch -> th ) )
3	2	a2i 11	. 2 |- ( ( ps -> ch ) -> ( ps -> th ) )
4	1, 3	syl 14	1 |- ( ph -> ( ps -> 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:  syl5com  29  syl6  33  syli  37  mpbidi  151  stdcn  848  con4biddc  858  jaddc  865  con1biddc  877  necon4addc  2445  necon4bddc  2446  necon4ddc  2447  necon1addc  2451  necon1bddc  2452  dmcosseq  4949  iss  5004  funopg  5304  snon0  7036  metrest  14949