Theorem ineqri 1637

Description: Inference from membership to intersection.

Hypothesis

Ref	Expression
ineqri.1	|- ((x e. A /\ x e. B) <-> x e. C)

Assertion

Ref	Expression
ineqri	|- (A i^i B) = C

Distinct variable group(s):   x,A   x,B   x,C

Proof of Theorem ineqri

Step	Hyp	Ref	Expression
1		elin 1635	. . 3 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
2		ineqri.1	. . 3 |- ((x e. A /\ x e. B) <-> x e. C)
3	1, 2	bitr 151	. 2 |- (x e. (A i^i B) <-> x e. C)
4	3	cleqri 1101	1 |- (A i^i B) = C

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092   i^i cin 1486

This theorem is referenced by:  inidm 1649  inass 1650  in0 1722  pwin 1915  dmres 2584

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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