Theorem intexrab 2737

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 2736	. 2 |- (E.x(x e. A /\ ph) <-> |^|{x | (x e. A /\ ph)} e. V)
2		df-rex 1653	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
3		df-rab 1655	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	inteqi 2541	. . 3 |- |^|{x e. A | ph} = |^|{x | (x e. A /\ ph)}
5	4	eleq1i 1540	. 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 960  E.wex 982  {cab 1466  E.wrex 1649  {crab 1651  Vcvv 1814  |^|cint 2537

This theorem is referenced by:  onintrab2 3020  cardval 4836  alephsuc 4877  clsval 7674  spanvalt 9294

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-13 971  ax-14 972  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  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-int 2538

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