Theorem sylan2d 294

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

Hypotheses

Ref	Expression
sylan2d.1	|- ( ph -> ( ps -> ch ) )
sylan2d.2	|- ( ph -> ( ( th /\ ch ) -> ta ) )

Assertion

Ref	Expression
sylan2d	|- ( ph -> ( ( th /\ ps ) -> ta ) )

Proof of Theorem sylan2d

Step	Hyp	Ref	Expression
1		sylan2d.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		sylan2d.2	. . . 4 |- ( ph -> ( ( th /\ ch ) -> ta ) )
3	2	ancomsd 269	. . 3 |- ( ph -> ( ( ch /\ th ) -> ta ) )
4	1, 3	syland 293	. 2 |- ( ph -> ( ( ps /\ th ) -> ta ) )
5	4	ancomsd 269	1 |- ( ph -> ( ( th /\ ps ) -> ta ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  syl2and  295  sylan2i  411  swopo  4451  prarloclemlo  7861  prodgt02  9183  prodge02  9185  infpnlem1  13138  ballotfilemfc0  13232  ballotfilemfcc  13233