Theorem wl-sb8t

Description: Substitution of variable in universal quantifier. Closed form of sb8 . (Contributed by Wolf Lammen, 27-Jul-2019)

	Ref	Expression
Assertion	wl-sb8t	$$⊢\forall xℲy\phi \to \left(\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall xℲy\phi$$
2		nfnf1	$$⊢ℲyℲy\phi$$
3	2	nfal	$$⊢Ⅎy\forall xℲy\phi$$
4		sp	$$⊢\forall xℲy\phi \to Ⅎy\phi$$
5		wl-nfs1t	$$⊢Ⅎy\phi \to Ⅎx\left[y/x\right]\phi$$
6	5	sps	$$⊢\forall xℲy\phi \to Ⅎx\left[y/x\right]\phi$$
7		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
8	7	a1i	$$⊢\forall xℲy\phi \to \left(x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)\right)$$
9	1 3 4 6 8	cbv2	$$⊢\forall xℲy\phi \to \left(\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi \right)$$