Theorem syld3an2 1321

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 1236	. . 3 |- ( ( ph /\ th /\ ch ) -> ps )
3		syld3an2.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com23 1236	. . 3 |- ( ( ph /\ th /\ ps ) -> ta )
5	2, 4	syld3an3 1319	. 2 |- ( ( ph /\ th /\ ch ) -> ta )
6	5	3com23 1236	1 |- ( ( ph /\ ch /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ w3a 1005

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 1007

This theorem is referenced by:  nppcan2  8469  nnncan  8473  nnncan2  8475  ltdivmul  9115  ledivmul  9116  ltdiv23  9131  lediv23  9132  pfxtrcfv  11340  dvdssub2  12476  dvdsgcdb  12664  lcmdvdsb  12736  ressabsg  13239  mulginvcom  13814  lspssp  14499  rpdivcxp  15722