Theorem syl2ani 688

Description: A syllogism inference. (Contributed by NM, 3-Aug-1999.)

Hypotheses

Ref	Expression
syl2ani.1	|- ( ph -> ch )
syl2ani.2	|- ( et -> th )
syl2ani.3	|- ( ps -> ( ( ch /\ th ) -> ta ) )

Assertion

Ref	Expression
syl2ani	|- ( ps -> ( ( ph /\ et ) -> ta ) )

Proof of Theorem syl2ani

Step	Hyp	Ref	Expression
1		syl2ani.1	. 2 |- ( ph -> ch )
2		syl2ani.2	. . 3 |- ( et -> th )
3		syl2ani.3	. . 3 |- ( ps -> ( ( ch /\ th ) -> ta ) )
4	2, 3	sylan2i 687	. 2 |- ( ps -> ( ( ch /\ et ) -> ta ) )
5	1, 4	sylani 686	1 |- ( ps -> ( ( ph /\ et ) -> ta ) )

Syntax hints:   -> wi 4   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  2mo  2551  frxp  7287  mapen  8124  fin1a2lem9  9230  coprmproddvdslem  15376  psss  17214  mgmidmo  17259  aannenlem1  24083  funtransport  32138  cgrxfr  32162  btwnxfr  32163  bj-cbv3tb  32711