Theorem sb8

Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8v . (Contributed by NM, 16-May-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Jim Kingdon, 15-Jan-2018) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb8.1	$$⊢Ⅎy\phi$$
Assertion	sb8	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb8.1	$$⊢Ⅎy\phi$$
2	1	nfs1	$$⊢Ⅎx\left[y/x\right]\phi$$
3		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
4	1 2 3	cbval	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$