Theorem rspc3ev

Description: 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012)

	Ref	Expression
Hypotheses	rspc3v.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
	rspc3v.2	$$⊢y=B\to \left(\chi \leftrightarrow \theta \right)$$
	rspc3v.3	$$⊢z=C\to \left(\theta \leftrightarrow \psi \right)$$
Assertion	rspc3ev	$$⊢\left(\left(A\in R\wedge B\in S\wedge C\in T\right)\wedge \psi \right)\to \exists x\in R\exists y\in S\exists z\in T\phi$$

Proof

Step	Hyp	Ref	Expression
1		rspc3v.1	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
2		rspc3v.2	$$⊢y=B\to \left(\chi \leftrightarrow \theta \right)$$
3		rspc3v.3	$$⊢z=C\to \left(\theta \leftrightarrow \psi \right)$$
4	1	rexbidv	$$⊢x=A\to \left(\exists z\in T\phi \leftrightarrow \exists z\in T\chi \right)$$
5	2	rexbidv	$$⊢y=B\to \left(\exists z\in T\chi \leftrightarrow \exists z\in T\theta \right)$$
6		simpl1	$$⊢\left(\left(A\in R\wedge B\in S\wedge C\in T\right)\wedge \psi \right)\to A\in R$$
7		simpl2	$$⊢\left(\left(A\in R\wedge B\in S\wedge C\in T\right)\wedge \psi \right)\to B\in S$$
8	3	rspcev	$$⊢\left(C\in T\wedge \psi \right)\to \exists z\in T\theta$$
9	8	3ad2antl3	$$⊢\left(\left(A\in R\wedge B\in S\wedge C\in T\right)\wedge \psi \right)\to \exists z\in T\theta$$
10	4 5 6 7 9	2rspcedvdw	$$⊢\left(\left(A\in R\wedge B\in S\wedge C\in T\right)\wedge \psi \right)\to \exists x\in R\exists y\in S\exists z\in T\phi$$