Theorem syl21anbrc 1213

Description: Syllogism inference. (Contributed by Peter Mazsa, 18-Sep-2022.)

Hypotheses

Ref	Expression
syl21anbrc.1	|- ( ph -> ps )
syl21anbrc.2	|- ( ph -> ch )
syl21anbrc.3	|- ( ph -> th )
syl21anbrc.4	|- ( ta <-> ( ( ps /\ ch ) /\ th ) )

Assertion

Ref	Expression
syl21anbrc	|- ( ph -> ta )

Proof of Theorem syl21anbrc

Step	Hyp	Ref	Expression
1		syl21anbrc.1	. . 3 |- ( ph -> ps )
2		syl21anbrc.2	. . 3 |- ( ph -> ch )
3		syl21anbrc.3	. . 3 |- ( ph -> th )
4	1, 2, 3	jca31 309	. 2 |- ( ph -> ( ( ps /\ ch ) /\ th ) )
5		syl21anbrc.4	. 2 |- ( ta <-> ( ( ps /\ ch ) /\ th ) )
6	4, 5	sylibr 134	1 |- ( ph -> ta )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

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:  idmhm  13776  resmhm2b  13796  mhmfmhm  13920  isghmd  14055  ghmmhm  14056  idghm  14062  isrhm2d  14472  subrgid  14531  issubrg2  14549  subsubrg  14553  aprap  14598  issubassa  15013