Theorem abv

Description: The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication ( bj-abv ) requires fewer axioms. (Contributed by BJ, 19-Mar-2021)

	Ref	Expression
Assertion	abv	$$⊢\left\{x|\phi \right\}=\mathrm{V}\leftrightarrow \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-clab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \left[y/x\right]\phi$$
2	1	albii	$$⊢\forall yy\in \left\{x|\phi \right\}\leftrightarrow \forall y\left[y/x\right]\phi$$
3		eqv	$$⊢\left\{x|\phi \right\}=\mathrm{V}\leftrightarrow \forall yy\in \left\{x|\phi \right\}$$
4		nfv	$$⊢Ⅎy\phi$$
5	4	sb8v	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$
6	2 3 5	3bitr4i	$$⊢\left\{x|\phi \right\}=\mathrm{V}\leftrightarrow \forall x\phi$$