Theorem syldanl 735

Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
syldanl	|- ( ( ( ph /\ ps ) /\ th ) -> ta )

Proof of Theorem syldanl

Step	Hyp	Ref	Expression
1		syldanl.1	. . . 4 |- ( ( ph /\ ps ) -> ch )
2	1	ex 450	. . 3 |- ( ph -> ( ps -> ch ) )
3	2	imdistani 726	. 2 |- ( ( ph /\ ps ) -> ( ph /\ ch ) )
4		syldanl.2	. 2 |- ( ( ( ph /\ ch ) /\ th ) -> ta )
5	3, 4	sylan 488	1 |- ( ( ( ph /\ ps ) /\ th ) -> 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:  trust  22033  submateq  29875  heibor1lem  33608  idlnegcl  33821  igenmin  33863  binomcxplemnotnn0  38555  vonioolem1  40894  vonicclem1  40897  smfsuplem1  41017  smflimsuplem4  41029