Theorem syld3an2 1325

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 1240	. . 3 |- ( ( ph /\ th /\ ch ) -> ps )
3		syld3an2.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com23 1240	. . 3 |- ( ( ph /\ th /\ ps ) -> ta )
5	2, 4	syld3an3 1323	. 2 |- ( ( ph /\ th /\ ch ) -> ta )
6	5	3com23 1240	1 |- ( ( ph /\ ch /\ 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:  nppcan2  8557  nnncan  8561  nnncan2  8563  ltdivmul  9206  ledivmul  9207  ltdiv23  9222  lediv23  9223  pfxtrcfv  11465  dvdssub2  12602  dvdsgcdb  12790  lcmdvdsb  12862  ressabsg  13430  mulginvcom  13950  lspssp  14740  rpdivcxp  16013