Theorem vtocldf

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypotheses	vtocld.1	$$⊢\phi \to A\in V$$
	vtocld.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
	vtocld.3	$$⊢\phi \to \psi$$
	vtocldf.4	$$⊢Ⅎx\phi$$
	vtocldf.5	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	vtocldf.6	$$⊢\phi \to Ⅎx\chi$$
Assertion	vtocldf	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		vtocld.1	$$⊢\phi \to A\in V$$
2		vtocld.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		vtocld.3	$$⊢\phi \to \psi$$
4		vtocldf.4	$$⊢Ⅎx\phi$$
5		vtocldf.5	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
6		vtocldf.6	$$⊢\phi \to Ⅎx\chi$$
7	2	ex	$$⊢\phi \to \left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)$$
8	4 7	alrimi	$$⊢\phi \to \forall x\left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)$$
9	4 3	alrimi	$$⊢\phi \to \forall x\psi$$
10		vtoclgft	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\chi \right)\wedge \left(\forall x\left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)\wedge \forall x\psi \right)\wedge A\in V\right)\to \chi$$
11	5 6 8 9 1 10	syl221anc	$$⊢\phi \to \chi$$