Theorem eqvincOLD 2388

Description: A variable introduction law for class equality.

Hypothesis

Ref	Expression
eqvinc.1	|- A e. _V

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem eqvincOLD

Step	Hyp	Ref	Expression
1		eqvinc.1	. . 3 |- A e. _V
2		eleq1 1957	. . 3 |- (A = B -> (A e. _V <-> B e. _V))
3	1, 2	mpbii 210	. 2 |- (A = B -> B e. _V)
4		visset 2295	. . . . 5 |- x e. _V
5		eleq1 1957	. . . . 5 |- (x = B -> (x e. _V <-> B e. _V))
6	4, 5	mpbii 210	. . . 4 |- (x = B -> B e. _V)
7	6	adantl 424	. . 3 |- ((x = A /\ x = B) -> B e. _V)
8	7	19.23aiv 1674	. 2 |- (E.x(x = A /\ x = B) -> B e. _V)
9		eqeq2 1893	. . 3 |- (z = B -> (A = z <-> A = B))
10		eqeq2 1893	. . . . 5 |- (z = B -> (x = z <-> x = B))
11	10	anbi2d 678	. . . 4 |- (z = B -> ((x = A /\ x = z) <-> (x = A /\ x = B)))
12	11	exbidv 1657	. . 3 |- (z = B -> (E.x(x = A /\ x = z) <-> E.x(x = A /\ x = B)))
13		eqeq1 1890	. . . 4 |- (y = A -> (y = z <-> A = z))
14		eqeq1 1890	. . . . . . 7 |- (y = A -> (y = x <-> A = x))
15		eqcom 1886	. . . . . . 7 |- (A = x <-> x = A)
16	14, 15	syl6bb 595	. . . . . 6 |- (y = A -> (y = x <-> x = A))
17	16	anbi1d 679	. . . . 5 |- (y = A -> ((y = x /\ x = z) <-> (x = A /\ x = z)))
18	17	exbidv 1657	. . . 4 |- (y = A -> (E.x(y = x /\ x = z) <-> E.x(x = A /\ x = z)))
19		equvin 1652	. . . 4 |- (y = z <-> E.x(y = x /\ x = z))
20	1, 13, 18, 19	vtoclb 2344	. . 3 |- (A = z <-> E.x(x = A /\ x = z))
21	9, 12, 20	vtoclbg 2347	. 2 |- (B e. _V -> (A = B <-> E.x(x = A /\ x = B)))
22	3, 8, 21	pm5.21nii 743	1 |- (A = B <-> E.x(x = A /\ x = B))

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

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  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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