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  847  con4biddc  857  jaddc  864  con1biddc  876  necon4addc  2417  necon4bddc  2418  necon4ddc  2419  necon1addc  2423  necon1bddc  2424  dmcosseq  4900  iss  4955  funopg  5252  snon0  6937  metrest  14091