Theorem syl2ani 660

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 659	. 2 |- ( ps -> ( ( ch /\ et ) -> ta ) )
5	1, 4	sylani 658	1 |- ( ps -> ( ( ph /\ et ) -> ta ) )

Syntax hints:   -> wi 4   /\ wa 370

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 188  df-an 372

This theorem is referenced by:  2mo  2347  frxp  6913  mapen  7738  fin1a2lem9  8838  coprmproddvdslem  14666  psss  16447  mgmidmo  16489  aannenlem1  23270  funtransport  30790  cgrxfr  30814  btwnxfr  30815  bj-cbv3tb  31258