Theorem syl2anr

Description: A double syllogism inference. For an implication-only version, see syl2imc . (Contributed by NM, 17-Sep-2013)

	Ref	Expression
Hypotheses	syl2an.1	|- ( ph -> ps )
	syl2an.2	|- ( ta -> ch )
	syl2an.3	|- ( ( ps /\ ch ) -> th )
Assertion	syl2anr	|- ( ( ta /\ ph ) -> th )

Proof

Step	Hyp	Ref	Expression
1		syl2an.1	|- ( ph -> ps )
2		syl2an.2	|- ( ta -> ch )
3		syl2an.3	|- ( ( ps /\ ch ) -> th )
4	1 2 3	syl2an	|- ( ( ph /\ ta ) -> th )
5	4	ancoms	|- ( ( ta /\ ph ) -> th )