Theorem barbarilem

Description: Lemma for barbari and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	barbarilem.min	|- E. x ph
	barbarilem.maj	|- A. x ( ph -> ps )
Assertion	barbarilem	|- E. x ( ph /\ ps )

Proof

Step	Hyp	Ref	Expression
1		barbarilem.min	|- E. x ph
2		barbarilem.maj	|- A. x ( ph -> ps )
3		exintr	|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
4	2 1 3	mp2	|- E. x ( ph /\ ps )