Theorem eqv 3178

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐴
2	1	eqvf 3177	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  abv  3179  dmi  5261  dfac10  8842  dfac10c  8843  dfac10b  8844  uniwun  9441  fnsingle  31196  bj-abv  32093  ttac  36621  nev  37081