Theorem eqvf 3204

Description: The universe contains every set. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
eqvf.1	|- F/_ x A

Assertion

Ref	Expression
eqvf	|- ( A = _V <-> A. x x e. A )

Proof of Theorem eqvf

Step	Hyp	Ref	Expression
1		eqvf.1	. . 3 |- F/_ x A
2		nfcv 2764	. . 3 |- F/_ x _V
3	1, 2	cleqf 2790	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
4		vex 3203	. . . 4 |- x e. _V
5	4	tbt 359	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
6	5	albii 1747	. 2 |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) )
7	3, 6	bitr4i 267	1 |- ( A = _V <-> A. x x e. A )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  eqv  3205