Theorem bj-hbsb3t

Description: A theorem close to a closed form of hbsb3 . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-hbsb3t	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		spsbim	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\forall y\phi \right)$$
2		hbsb2a	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left[y/x\right]\phi$$
3	1 2	syl6	$$⊢\forall x\left(\phi \to \forall y\phi \right)\to \left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$