Theorem vtocl3g

Description: Implicit substitution of a class for a setvar variable. Version of vtocl3gf with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024)

	Ref	Expression
Hypotheses	vtocl3g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	vtocl3g.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
	vtocl3g.3	$$⊢z=C\to \left(\chi \leftrightarrow \theta \right)$$
	vtocl3g.4	$$⊢\phi$$
Assertion	vtocl3g	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \theta$$

Proof

Step	Hyp	Ref	Expression
1		vtocl3g.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		vtocl3g.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
3		vtocl3g.3	$$⊢z=C\to \left(\chi \leftrightarrow \theta \right)$$
4		vtocl3g.4	$$⊢\phi$$
5		elex	$$⊢A\in V\to A\in \mathrm{V}$$
6	2	imbi2d	$$⊢y=B\to \left(\left(A\in \mathrm{V}\to \psi \right)\leftrightarrow \left(A\in \mathrm{V}\to \chi \right)\right)$$
7	3	imbi2d	$$⊢z=C\to \left(\left(A\in \mathrm{V}\to \chi \right)\leftrightarrow \left(A\in \mathrm{V}\to \theta \right)\right)$$
8	1 4	vtoclg	$$⊢A\in \mathrm{V}\to \psi$$
9	6 7 8	vtocl2g	$$⊢\left(B\in W\wedge C\in X\right)\to \left(A\in \mathrm{V}\to \theta \right)$$
10	5 9	mpan9	$$⊢\left(A\in V\wedge \left(B\in W\wedge C\in X\right)\right)\to \theta$$
11	10	3impb	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \theta$$