Theorem syld3an2 1216

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 1144	. . 3 |- ( ( ph /\ th /\ ch ) -> ps )
3		syld3an2.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com23 1144	. . 3 |- ( ( ph /\ th /\ ps ) -> ta )
5	2, 4	syld3an3 1214	. 2 |- ( ( ph /\ th /\ ch ) -> ta )
6	5	3com23 1144	1 |- ( ( ph /\ ch /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ w3a 919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 921

This theorem is referenced by:  nppcan2  7339  nnncan  7343  nnncan2  7345  ltdivmul  7954  ledivmul  7955  ltdiv23  7970  lediv23  7971  dvdssub2  10237  dvdsgcdb  10402  lcmdvdsb  10466