Theorem syl2and 295

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 294	. 2 |- ( ph -> ( ( ch /\ th ) -> et ) )
5	1, 4	syland 293	1 |- ( ph -> ( ( ps /\ th ) -> et ) )

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:  anim12d  335  recexprlem1ssl  8000  recexprlem1ssu  8001  xle2add  10283  fzen  10449  bezoutlembi  12784  rpmulgcd2  12875  pcqmul  13084  mpodvdsmulf1o  16110  2sqlem8a  16253  uspgr2wlkeq  16618