Theorem eqv 2292

Description: The universe contains every set.

Assertion

Ref	Expression
eqv	|- (A = V <-> A.x x e. A)

Distinct variable group:   x,A

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 1469	. 2 |- (A = V <-> A.x(x e. A <-> x e. V))
2		visset 1810	. . . 4 |- x e. V
3	2	tbt 719	. . 3 |- (x e. A <-> (x e. A <-> x e. V))
4	3	albii 998	. 2 |- (A.x x e. A <-> A.x(x e. A <-> x e. V))
5	1, 4	bitr4 176	1 |- (A = V <-> A.x x e. A)

Syntax hints:   <-> wb 146  A.wal 953   = wceq 955   e. wcel 957  Vcvv 1808

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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