Theorem eqvf 3461

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

Hypothesis

Ref	Expression
eqvf.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
eqvf	⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Proof of Theorem eqvf

Step	Hyp	Ref	Expression
1		eqvf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2892	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	cleqf 2921	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
5	4	tbt 369	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
6	5	albii 1819	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
7	3, 6	bitr4i 278	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2877  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-v 3452

This theorem is referenced by: (None)