Theorem eqv 3448

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2114, ax-11 2128, ax-13 2346. (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 2791	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
2		vex 3443	. . . 4 ⊢ 𝑥 ∈ V
3	2	tbt 371	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4	3	albii 1805	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
5	1, 4	bitr4i 279	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 207  ∀wal 1523   = wceq 1525   ∈ wcel 2083  Vcvv 3440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-v 3442

This theorem is referenced by:  abv  3450  dmi  5684  dfac10  9416  dfac10c  9417  dfac10b  9418  uniwun  10015  fnsingle  32991  bj-abv  33800  ttac  39139  nev  39621