Theorem eqv 3461

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2178, ax-11 2194, ax-13 2402. (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 2754	. 2 ⊢ (𝐴 = V ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
2		vex 3455	. . . 4 ⊢ 𝑥 ∈ V
3	2	tbt 372	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
4	3	albii 1852	. 2 ⊢ (∀𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ V))
5	1, 4	bitr4i 281	1 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2145  Vcvv 3451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453

This theorem is used by:  abvALT  3464  dmi  5903  dmep  5905  dfac10  10209  dfac10c  10210  dfac10b  10211  uniwun  10818  onvf1odlem1  35865  onvf1odlem4  35868  fnsingle  36661  bj-abvALT  37799  ttac  44022  nev  44755