Theorem syl2an3an 1288

Description: syl3an 1270 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)

Hypotheses

Ref	Expression
syl2an3an.1	|- ( ph -> ps )
syl2an3an.2	|- ( ph -> ch )
syl2an3an.3	|- ( th -> ta )
syl2an3an.4	|- ( ( ps /\ ch /\ ta ) -> et )

Assertion

Ref	Expression
syl2an3an	|- ( ( ph /\ th ) -> et )

Proof of Theorem syl2an3an

Step	Hyp	Ref	Expression
1		syl2an3an.1	. . 3 |- ( ph -> ps )
2		syl2an3an.2	. . 3 |- ( ph -> ch )
3		syl2an3an.3	. . 3 |- ( th -> ta )
4		syl2an3an.4	. . 3 |- ( ( ps /\ ch /\ ta ) -> et )
5	1, 2, 3, 4	syl3an 1270	. 2 |- ( ( ph /\ ph /\ th ) -> et )
6	5	3anidm12 1285	1 |- ( ( ph /\ th ) -> et )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968

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 970

This theorem is referenced by:  expcnvap0  11443  efexp  11623  cncongr1  12035  uptx  12914  logbgcd1irr  13525