Theorem sb8v

Description: Substitution of variable in universal quantifier. Version of sb8f with a disjoint variable condition replacing the nonfree hypothesis F/ y ph , not requiring ax-12 . (Contributed by SN, 5-Dec-2024)

	Ref	Expression
Assertion	sb8v	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb6	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
2	1	albii	$$⊢\forall y\left[y/x\right]\phi \leftrightarrow \forall y\forall x\left(x=y\to \phi \right)$$
3		alcom	$$⊢\forall y\forall x\left(x=y\to \phi \right)\leftrightarrow \forall x\forall y\left(x=y\to \phi \right)$$
4		equcom	$$⊢x=y\leftrightarrow y=x$$
5	4	imbi1i	$$⊢\left(x=y\to \phi \right)\leftrightarrow \left(y=x\to \phi \right)$$
6	5	albii	$$⊢\forall y\left(x=y\to \phi \right)\leftrightarrow \forall y\left(y=x\to \phi \right)$$
7		equsv	$$⊢\forall y\left(y=x\to \phi \right)\leftrightarrow \phi$$
8	6 7	bitri	$$⊢\forall y\left(x=y\to \phi \right)\leftrightarrow \phi$$
9	8	albii	$$⊢\forall x\forall y\left(x=y\to \phi \right)\leftrightarrow \forall x\phi$$
10	2 3 9	3bitrri	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$