Theorem eqvf

Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021)

	Ref	Expression
Hypothesis	eqvf.1	⊢ Ⅎ 𝑥 𝐴
Assertion	eqvf	⊢ ( 𝐴 = V ↔ ∀ 𝑥 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqvf.1	⊢ Ⅎ 𝑥 𝐴
2		nfcv	⊢ Ⅎ 𝑥 V
3	1 2	cleqf	⊢ ( 𝐴 = V ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V ) )
4		vex	⊢ 𝑥 ∈ V
5	4	tbt	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V ) )
6	5	albii	⊢ ( ∀ 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V ) )
7	3 6	bitr4i	⊢ ( 𝐴 = V ↔ ∀ 𝑥 𝑥 ∈ 𝐴 )