Theorem bj-abv

Description: The class of sets verifying a tautology is the universal class. (Contributed by BJ, 24-Jul-2019) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢\forall x\phi \to \forall y\forall x\phi$$
2		vexwt	$$⊢\forall x\phi \to y\in \left\{x|\phi \right\}$$
3	1 2	alrimih	$$⊢\forall x\phi \to \forall yy\in \left\{x|\phi \right\}$$
4		eqv	$$⊢\left\{x|\phi \right\}=\mathrm{V}\leftrightarrow \forall yy\in \left\{x|\phi \right\}$$
5	3 4	sylibr	$$⊢\forall x\phi \to \left\{x|\phi \right\}=\mathrm{V}$$