Theorem eqvinc 2387

Description: A variable introduction law for class equality. (The proof was shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
eqvinc.1	|- A e. _V

Assertion

Ref	Expression
eqvinc	|- (A = B <-> E.x(x = A /\ x = B))

Distinct variable groups:   x,A   x,B

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. . . . . 6 |- A e. _V
2	1	isseti 2297	. . . . 5 |- E.x x = A
3		ax-1 4	. . . . . . 7 |- (x = A -> (A = B -> x = A))
4		eqtr 1904	. . . . . . . 8 |- ((x = A /\ A = B) -> x = B)
5	4	ex 402	. . . . . . 7 |- (x = A -> (A = B -> x = B))
6	3, 5	jca 310	. . . . . 6 |- (x = A -> ((A = B -> x = A) /\ (A = B -> x = B)))
7	6	eximi 1387	. . . . 5 |- (E.x x = A -> E.x((A = B -> x = A) /\ (A = B -> x = B)))
8	2, 7	ax-mp 7	. . . 4 |- E.x((A = B -> x = A) /\ (A = B -> x = B))
9		pm3.43 664	. . . . 5 |- (((A = B -> x = A) /\ (A = B -> x = B)) -> (A = B -> (x = A /\ x = B)))
10	9	eximi 1387	. . . 4 |- (E.x((A = B -> x = A) /\ (A = B -> x = B)) -> E.x(A = B -> (x = A /\ x = B)))
11	8, 10	ax-mp 7	. . 3 |- E.x(A = B -> (x = A /\ x = B))
12	11	19.37aiv 1684	. 2 |- (A = B -> E.x(x = A /\ x = B))
13		eqtr2 1905	. . 3 |- ((x = A /\ x = B) -> A = B)
14	13	19.23aiv 1674	. 2 |- (E.x(x = A /\ x = B) -> A = B)
15	12, 14	impbii 174	1 |- (A = B <-> E.x(x = A /\ x = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  eqvincf 2389  moi2 2435  moi 2436  opabidOLD 3558  tfindsg 3944  findsg 3980  dff13 4850  dfoprab5sf 5058  indpi 6186

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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