Theorem syld3an2 1280

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 1204	. . 3 |- ( ( ph /\ th /\ ch ) -> ps )
3		syld3an2.2	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
4	3	3com23 1204	. . 3 |- ( ( ph /\ th /\ ps ) -> ta )
5	2, 4	syld3an3 1278	. 2 |- ( ( ph /\ th /\ ch ) -> ta )
6	5	3com23 1204	1 |- ( ( ph /\ ch /\ th ) -> ta )

Syntax hints:   -> wi 4   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  nppcan2  8137  nnncan  8141  nnncan2  8143  ltdivmul  8779  ledivmul  8780  ltdiv23  8795  lediv23  8796  dvdssub2  11784  dvdsgcdb  11955  lcmdvdsb  12025  rpdivcxp  13547