Theorem intexrab 2727

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 2726	. 2 |- (E.x(x e. A /\ ph) <-> |^|{x | (x e. A /\ ph)} e. V)
2		df-rex 1647	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
3		df-rab 1649	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	inteqi 2532	. . 3 |- |^|{x e. A | ph} = |^|{x | (x e. A /\ ph)}
5	4	eleq1i 1534	. 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 956  E.wex 978  {cab 1461  E.wrex 1643  {crab 1645  Vcvv 1807  |^|cint 2528

This theorem is referenced by:  onintrab2 3009  cardval 4806  alephsuc 4846  clsval 7627  spanvalt 9237

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-int 2529

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