Theorem sbcal

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	sbcal	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\phi \leftrightarrow \forall x\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\phi \to A\in \mathrm{V}$$
2		sbcex	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$
3	2	sps	$$⊢\forall x\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$
4		dfsbcq2	$$⊢z=A\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\phi \right)$$
5		dfsbcq2	$$⊢z=A\to \left(\left[z/y\right]\phi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi \right)$$
6	5	albidv	$$⊢z=A\to \left(\forall x\left[z/y\right]\phi \leftrightarrow \forall x\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi \right)$$
7		sbal	$$⊢\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi$$
8	4 6 7	vtoclbg	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\phi \leftrightarrow \forall x\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi \right)$$
9	1 3 8	pm5.21nii	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\phi \leftrightarrow \forall x\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$