Theorem sbalv

Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypothesis	sbalv.1	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
Assertion	sbalv	$$⊢\left[y/x\right]\forall z\phi \leftrightarrow \forall z\psi$$

Proof

Step	Hyp	Ref	Expression
1		sbalv.1	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
2		sbal	$$⊢\left[y/x\right]\forall z\phi \leftrightarrow \forall z\left[y/x\right]\phi$$
3	1	albii	$$⊢\forall z\left[y/x\right]\phi \leftrightarrow \forall z\psi$$
4	2 3	bitri	$$⊢\left[y/x\right]\forall z\phi \leftrightarrow \forall z\psi$$