Theorem eqv 3490

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2141, ax-11 2157, ax-13 2377. (Revised by BJ, 10-Aug-2022.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 2730	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
2		vex 3484	. . . 4 ⊢ 𝑥 ∈ V
3	2	tbt 369	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4	3	albii 1819	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
5	1, 4	bitr4i 278	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2108  Vcvv 3480

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482

This theorem is referenced by:  abvALT  3493  dmi  5932  dmep  5934  dfac10  10178  dfac10c  10179  dfac10b  10180  uniwun  10780  fnsingle  35920  bj-abvALT  36908  ttac  43048  nev  43783