Theorem eqv 3446

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2144, ax-11 2160, ax-13 2372. (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 2724	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
2		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
3	2	tbt 369	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4	3	albii 1820	. 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 1539   = wceq 1541   ∈ wcel 2111  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438

This theorem is referenced by:  abvALT  3449  dmi  5860  dmep  5862  dfac10  10029  dfac10c  10030  dfac10b  10031  uniwun  10631  onvf1odlem1  35147  onvf1odlem4  35150  fnsingle  35961  bj-abvALT  36949  ttac  43077  nev  43811