Theorem elqsi 4291

Description: Membership in a quotient set.

Assertion

Ref	Expression
elqsi	|- (B e. (A/.R) -> E.x(x e. A /\ B = [x]R))

Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem elqsi

Step	Hyp	Ref	Expression
1		eqeq1 1481	. . . 4 |- (y = B -> (y = [x]R <-> B = [x]R))
2	1	anbi2d 616	. . 3 |- (y = B -> ((x e. A /\ y = [x]R) <-> (x e. A /\ B = [x]R)))
3	2	exbidv 1279	. 2 |- (y = B -> (E.x(x e. A /\ y = [x]R) <-> E.x(x e. A /\ B = [x]R)))
4		visset 1813	. . . 4 |- y e. V
5	4	elqs 4290	. . 3 |- (y e. (A/.R) <-> E.x(x e. A /\ y = [x]R))
6	5	biimp 151	. 2 |- (y e. (A/.R) -> E.x(x e. A /\ y = [x]R))
7	3, 6	vtoclga 1852	1 |- (B e. (A/.R) -> E.x(x e. A /\ B = [x]R))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  [cec 4259  /.cqs 4260

This theorem is referenced by:  0nelqs 4298  ectocl 4302  ecoptocl 4303

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812  df-qs 4266

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