Theorem syld3an3 1262

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

Hypotheses

Ref	Expression
syld3an3.1	|- ( ( ph /\ ps /\ ch ) -> th )
syld3an3.2	|- ( ( ph /\ ps /\ th ) -> ta )

Assertion

Ref	Expression
syld3an3	|- ( ( ph /\ ps /\ ch ) -> ta )

Proof of Theorem syld3an3

Step	Hyp	Ref	Expression
1		simp1 982	. 2 |- ( ( ph /\ ps /\ ch ) -> ph )
2		simp2 983	. 2 |- ( ( ph /\ ps /\ ch ) -> ps )
3		syld3an3.1	. 2 |- ( ( ph /\ ps /\ ch ) -> th )
4		syld3an3.2	. 2 |- ( ( ph /\ ps /\ th ) -> ta )
5	1, 2, 3, 4	syl3anc 1217	1 |- ( ( ph /\ ps /\ ch ) -> ta )

Syntax hints:   -> wi 4   /\ w3a 963

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 965

This theorem is referenced by:  syld3an1  1263  syld3an2  1264  brelrng  4778  moriotass  5766  nnncan1  8022  lediv1  8651  modqval  10128  modqvalr  10129  modqcl  10130  flqpmodeq  10131  modq0  10133  modqge0  10136  modqlt  10137  modqdiffl  10139  modqdifz  10140  modqvalp1  10147  exp3val  10326  bcval4  10530  dvdsmultr1  11567  dvdssub2  11571  divalglemeuneg  11656  ndvdsadd  11664  basgen2  12289  opnneiss  12366  cnpf2  12415  sincosq1lem  12954