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-1 5  ax-2 6  ax-mp 7

This theorem is referenced by:  syl5com  29  syl6  33  syli  37  mpbidi  149  con4biddc  788  jaddc  795  con1biddc  804  necon4addc  2319  necon4bddc  2320  necon4ddc  2321  necon1addc  2325  necon1bddc  2326  dmcosseq  4651  iss  4704  funopg  4984  snon0  6477