Theorem syl22anbrc

Description: Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025)

	Ref	Expression
Hypotheses	syl22anbrc.1	|- ( ph -> ps )
	syl22anbrc.2	|- ( ph -> ch )
	syl22anbrc.3	|- ( ph -> th )
	syl22anbrc.4	|- ( ph -> ta )
	syl22anbrc.5	|- ( et <-> ( ( ps /\ ch ) /\ ( th /\ ta ) ) )
Assertion	syl22anbrc	|- ( ph -> et )

Proof

Step	Hyp	Ref	Expression
1		syl22anbrc.1	|- ( ph -> ps )
2		syl22anbrc.2	|- ( ph -> ch )
3		syl22anbrc.3	|- ( ph -> th )
4		syl22anbrc.4	|- ( ph -> ta )
5		syl22anbrc.5	|- ( et <-> ( ( ps /\ ch ) /\ ( th /\ ta ) ) )
6	3 4	jca	|- ( ph -> ( th /\ ta ) )
7	1 2 6 5	syl21anbrc	|- ( ph -> et )