Theorem syl2and 289

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Hypotheses

Ref	Expression
syl2and.1	|- ( ph -> ( ps -> ch ) )
syl2and.2	|- ( ph -> ( th -> ta ) )
syl2and.3	|- ( ph -> ( ( ch /\ ta ) -> et ) )

Assertion

Ref	Expression
syl2and	|- ( ph -> ( ( ps /\ th ) -> et ) )

Proof of Theorem syl2and

Step	Hyp	Ref	Expression
1		syl2and.1	. 2 |- ( ph -> ( ps -> ch ) )
2		syl2and.2	. . 3 |- ( ph -> ( th -> ta ) )
3		syl2and.3	. . 3 |- ( ph -> ( ( ch /\ ta ) -> et ) )
4	2, 3	sylan2d 288	. 2 |- ( ph -> ( ( ch /\ th ) -> et ) )
5	1, 4	syland 287	1 |- ( ph -> ( ( ps /\ th ) -> et ) )

Syntax hints:   -> wi 4   /\ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  anim12d  328  recexprlem1ssl  7182  recexprlem1ssu  7183  fzen  9447  bezoutlembi  11259  rpmulgcd2  11342