Theorem syld3an2 1320

Description: A syllogism inference. (Contributed by NM, 20-May-2007.)

Hypotheses

Ref	Expression
syld3an2.1	|- ( ( ph /\ ch /\ th ) -> ps )
syld3an2.2	|- ( ( ph /\ ps /\ th ) -> ta )

Assertion

Ref	Expression
syld3an2	|- ( ( ph /\ ch /\ th ) -> ta )

Proof of Theorem syld3an2

Step	Hyp	Ref	Expression
1		syld3an2.1	. . . 4 |- ( ( ph /\ ch /\ th ) -> ps )
2	1	3com23 1235	. . 3 |- ( ( ph /\ th /\ ch ) -> ps )
3		syld3an2.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com23 1235	. . 3 |- ( ( ph /\ th /\ ps ) -> ta )
5	2, 4	syld3an3 1318	. 2 |- ( ( ph /\ th /\ ch ) -> ta )
6	5	3com23 1235	1 |- ( ( ph /\ ch /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ w3a 1004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  nppcan2  8409  nnncan  8413  nnncan2  8415  ltdivmul  9055  ledivmul  9056  ltdiv23  9071  lediv23  9072  pfxtrcfv  11273  dvdssub2  12395  dvdsgcdb  12583  lcmdvdsb  12655  ressabsg  13158  mulginvcom  13733  lspssp  14416  rpdivcxp  15634