Theorem syld3an1 1324

Description: A syllogism inference. (Contributed by NM, 7-Jul-2008.)

Hypotheses

Ref	Expression
syld3an1.1	|- ( ( ch /\ ps /\ th ) -> ph )
syld3an1.2	|- ( ( ph /\ ps /\ th ) -> ta )

Assertion

Ref	Expression
syld3an1	|- ( ( ch /\ ps /\ th ) -> ta )

Proof of Theorem syld3an1

Step	Hyp	Ref	Expression
1		syld3an1.1	. . . 4 |- ( ( ch /\ ps /\ th ) -> ph )
2	1	3com13 1239	. . 3 |- ( ( th /\ ps /\ ch ) -> ph )
3		syld3an1.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com13 1239	. . 3 |- ( ( th /\ ps /\ ph ) -> ta )
5	2, 4	syld3an3 1323	. 2 |- ( ( th /\ ps /\ ch ) -> ta )
6	5	3com13 1239	1 |- ( ( ch /\ ps /\ th ) -> ta )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  tfrcllembacc  6626  npncan  8547  nnpcan  8549  ppncan  8568  muldivdirap  9037  div2negap  9065  ltmuldiv  9204  mulqmod0  10767  gcdaddm  12761  zndvds  14984