Theorem rspc3v

Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005)

	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	rspc3v	$$⊢\left(A\in R\wedge B\in S\wedge C\in T\right)\to \left(\forall x\in R\forall y\in S\forall z\in T\phi \to \psi \right)$$

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	ralbidv	$$⊢x=A\to \left(\forall z\in T\phi \leftrightarrow \forall z\in T\chi \right)$$
5	2	ralbidv	$$⊢y=B\to \left(\forall z\in T\chi \leftrightarrow \forall z\in T\theta \right)$$
6	4 5	rspc2v	$$⊢\left(A\in R\wedge B\in S\right)\to \left(\forall x\in R\forall y\in S\forall z\in T\phi \to \forall z\in T\theta \right)$$
7	3	rspcv	$$⊢C\in T\to \left(\forall z\in T\theta \to \psi \right)$$
8	6 7	sylan9	$$⊢\left(\left(A\in R\wedge B\in S\right)\wedge C\in T\right)\to \left(\forall x\in R\forall y\in S\forall z\in T\phi \to \psi \right)$$
9	8	3impa	$$⊢\left(A\in R\wedge B\in S\wedge C\in T\right)\to \left(\forall x\in R\forall y\in S\forall z\in T\phi \to \psi \right)$$