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 ) )

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  syl5com  29  syl6  33  syli  37  mpbidi  151  stdcn  859  con4biddc  869  jaddc  876  con1biddc  888  necon4addc  2490  necon4bddc  2491  necon4ddc  2492  necon1addc  2496  necon1bddc  2497  dmcosseq  5056  iss  5112  funopg  5414  snon0  7253  gsumcom2fi  14275  metrest  15732