Theorem syl21anbrc 1184

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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  idmhm  13101  resmhm2b  13121  mhmfmhm  13247  isghmd  13382  ghmmhm  13383  idghm  13389  isrhm2d  13721  subrgid  13779  issubrg2  13797  subsubrg  13801  aprap  13842