Theorem rspc2ev

Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999)

	Ref	Expression
Hypotheses	rspc2v.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
	rspc2v.2	$$⊢y=B\to \left(\chi \leftrightarrow \psi \right)$$
Assertion	rspc2ev	$$⊢\left(A\in C\wedge B\in D\wedge \psi \right)\to \exists x\in C\exists y\in D\phi$$

Proof

Step	Hyp	Ref	Expression
1		rspc2v.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
2		rspc2v.2	$$⊢y=B\to \left(\chi \leftrightarrow \psi \right)$$
3	1	rexbidv	$$⊢x=A\to \left(\exists y\in D\phi \leftrightarrow \exists y\in D\chi \right)$$
4		simp1	$$⊢\left(A\in C\wedge B\in D\wedge \psi \right)\to A\in C$$
5	2	rspcev	$$⊢\left(B\in D\wedge \psi \right)\to \exists y\in D\chi$$
6	5	3adant1	$$⊢\left(A\in C\wedge B\in D\wedge \psi \right)\to \exists y\in D\chi$$
7	3 4 6	rspcedvdw	$$⊢\left(A\in C\wedge B\in D\wedge \psi \right)\to \exists x\in C\exists y\in D\phi$$