Theorem rspcime

Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	rspcime.1	|- ( ( ph /\ x = A ) -> ps )
	rspcime.2	|- ( ph -> A e. B )
Assertion	rspcime	|- ( ph -> E. x e. B ps )

Proof

Step	Hyp	Ref	Expression
1		rspcime.1	|- ( ( ph /\ x = A ) -> ps )
2		rspcime.2	|- ( ph -> A e. B )
3		simpl	|- ( ( ph /\ x = A ) -> ph )
4	1 3	2thd	|- ( ( ph /\ x = A ) -> ( ps <-> ph ) )
5		id	|- ( ph -> ph )
6	2 4 5	rspcedvd	|- ( ph -> E. x e. B ps )