Theorem ineqri 2212

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 groups:   x,A   x,B   x,C

Proof of Theorem ineqri

Step	Hyp	Ref	Expression
1		elin 2210	. . 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 173	. 2 |- (x e. (A i^i B) <-> x e. C)
4	3	eqriv 1477	1 |- (A i^i B) = C

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   i^i cin 2049

This theorem is referenced by:  inidm 2225  inass 2226  in0 2302  pwin 2831  dmres 3386  pjhmopidm 10105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054

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