Theorem syl2an3an 1339

Description: syl3an 1320 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 1320	. 2 |- ( ( ph /\ ph /\ th ) -> et )
6	5	3anidm12 1336	1 |- ( ( ph /\ th ) -> et )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009

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  df-3an 1011

This theorem is used by:  prfidceq  7235  ccatass  11378  ccatpfx  11475  swrdswrd  11479  expcnvap0  12271  efexp  12451  cncongr1  12883  uptx  15377  logbgcd1irr  16075  gausslemma2dlem2  16193  wlkeq  16607