Theorem eqvf 3397

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 2947	. . 3 ⊢ Ⅎ𝑥V
3	1, 2	cleqf 2973	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4		vex 3393	. . . 4 ⊢ 𝑥 ∈ V
5	4	tbt 360	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
6	5	albii 1907	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
7	3, 6	bitr4i 269	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 197  ∀wal 1635   = wceq 1637   ∈ wcel 2158  Ⅎwnfc 2934  Vcvv 3390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-v 3392

This theorem is referenced by:  eqvOLD  3398