Theorem intexrab 2732

Description: The intersection of a non-empty restricted class abstraction exists.

Assertion

Ref	Expression
intexrab	|- (E.x e. A ph <-> |^|{x e. A | ph} e. V)

Proof of Theorem intexrab

Step	Hyp	Ref	Expression
1		intexab 2731	. 2 |- (E.x(x e. A /\ ph) <-> |^|{x | (x e. A /\ ph)} e. V)
2		df-rex 1650	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
3		df-rab 1652	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	inteqi 2537	. . 3 |- |^|{x e. A | ph} = |^|{x | (x e. A /\ ph)}
5	4	eleq1i 1537	. 2 |- (|^|{x e. A | ph} e. V <-> |^|{x | (x e. A /\ ph)} e. V)
6	1, 2, 5	3bitr4 183	1 |- (E.x e. A ph <-> |^|{x e. A | ph} e. V)

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  {crab 1648  Vcvv 1811  |^|cint 2533

This theorem is referenced by:  onintrab2 3014  cardval 4826  alephsuc 4866  clsval 7677  spanvalt 9299

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-13 969  ax-14 970  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  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-int 2534

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