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  854  con4biddc  864  jaddc  871  con1biddc  883  necon4addc  2472  necon4bddc  2473  necon4ddc  2474  necon1addc  2478  necon1bddc  2479  dmcosseq  5004  iss  5059  funopg  5360  snon0  7133  metrest  15229