Theorem eqvinc 1874

Description: A variable introduction law for class equality.

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	. . 3 |- A e. V
2		eleq1 1526	. . 3 |- (A = B -> (A e. V <-> B e. V))
3	1, 2	mpbii 193	. 2 |- (A = B -> B e. V)
4		visset 1804	. . . . 5 |- x e. V
5		eleq1 1526	. . . . 5 |- (x = B -> (x e. V <-> B e. V))
6	4, 5	mpbii 193	. . . 4 |- (x = B -> B e. V)
7	6	adantl 388	. . 3 |- ((x = A /\ x = B) -> B e. V)
8	7	19.23aiv 1290	. 2 |- (E.x(x = A /\ x = B) -> B e. V)
9		eqeq2 1476	. . 3 |- (z = B -> (A = z <-> A = B))
10		eqeq2 1476	. . . . 5 |- (z = B -> (x = z <-> x = B))
11	10	anbi2d 614	. . . 4 |- (z = B -> ((x = A /\ x = z) <-> (x = A /\ x = B)))
12	11	exbidv 1274	. . 3 |- (z = B -> (E.x(x = A /\ x = z) <-> E.x(x = A /\ x = B)))
13		eqeq1 1473	. . . 4 |- (y = A -> (y = z <-> A = z))
14		eqeq1 1473	. . . . . . 7 |- (y = A -> (y = x <-> A = x))
15		eqcom 1469	. . . . . . 7 |- (A = x <-> x = A)
16	14, 15	syl6bb 534	. . . . . 6 |- (y = A -> (y = x <-> x = A))
17	16	anbi1d 615	. . . . 5 |- (y = A -> ((y = x /\ x = z) <-> (x = A /\ x = z)))
18	17	exbidv 1274	. . . 4 |- (y = A -> (E.x(y = x /\ x = z) <-> E.x(x = A /\ x = z)))
19		equvin 1270	. . . 4 |- (y = z <-> E.x(y = x /\ x = z))
20	1, 13, 18, 19	vtoclb 1836	. . 3 |- (A = z <-> E.x(x = A /\ x = z))
21	9, 12, 20	vtoclbg 1839	. 2 |- (B e. V -> (A = B <-> E.x(x = A /\ x = B)))
22	3, 8, 21	pm5.21nii 677	1 |- (A = B <-> E.x(x = A /\ x = B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802

This theorem is referenced by:  eqvincf 1875  moi 1915  opabid 2799  findsg 3147  tfindsg 3152  ralxpf 3210  f1fv 3859  oprabval2gf 4011  indpi 5006  fsum1f 6945  fsump1f 6949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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