Theorem syl2an23an 1277

Description: Deduction related to syl3an 1258 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)

Hypotheses

Ref	Expression
syl2an23an.1	|- ( ph -> ps )
syl2an23an.2	|- ( ph -> ch )
syl2an23an.3	|- ( ( th /\ ph ) -> ta )
syl2an23an.4	|- ( ( ps /\ ch /\ ta ) -> et )

Assertion

Ref	Expression
syl2an23an	|- ( ( th /\ ph ) -> et )

Proof of Theorem syl2an23an

Step	Hyp	Ref	Expression
1		syl2an23an.3	. . 3 |- ( ( th /\ ph ) -> ta )
2		syl2an23an.1	. . . 4 |- ( ph -> ps )
3		syl2an23an.2	. . . 4 |- ( ph -> ch )
4		syl2an23an.4	. . . . 5 |- ( ( ps /\ ch /\ ta ) -> et )
5	4	3exp 1180	. . . 4 |- ( ps -> ( ch -> ( ta -> et ) ) )
6	2, 3, 5	sylc 62	. . 3 |- ( ph -> ( ta -> et ) )
7	1, 6	syl5 32	. 2 |- ( ph -> ( ( th /\ ph ) -> et ) )
8	7	anabsi7 570	1 |- ( ( th /\ ph ) -> et )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by:  fsum3ser  11166