Theorem intexrab 1988

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 1987	. 2 |- (E.x(x e. A /\ ph) <-> |^|{x | (x e. A /\ ph)} e. V)
2		df-rex 1206	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
3		df-rab 1208	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	inteqi 1969	. . 3 |- |^|{x e. A | ph} = |^|{x | (x e. A /\ ph)}
5	4	eleq1i 1152	. 2 |- (|^|{x e. A | ph} e. V <-> |^|{x | (x e. A /\ ph)} e. V)
6	1, 2, 5	3bitr4 158	1 |- (E.x e. A ph <-> |^|{x e. A | ph} e. V)

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678  {cab 1090   e. wcel 1092  E.wrex 1202  {crab 1204  Vcvv 1348  |^|cint 1965

This theorem is referenced by:  onintrab2 2269  cardval 3633  alephsuc 3672  spanvalt 5300

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-int 1966

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