Theorem intexrabim 4171

Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intexrabim	|- ( E. x e. A ph -> |^| { x e. A | ph } e. _V )

Proof of Theorem intexrabim

Step	Hyp	Ref	Expression
1		intexabim 4170	. 2 |- ( E. x ( x e. A /\ ph ) -> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2474	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2477	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 3863	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2255	. 2 |- ( |^| { x e. A | ph } e. _V <-> |^| { x | ( x e. A /\ ph ) } e. _V )
6	1, 2, 5	3imtr4i 201	1 |- ( E. x e. A ph -> |^| { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1503   e. wcel 2160   {cab 2175   E.wrex 2469   {crab 2472   _Vcvv 2752   |^|cint 3859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-int 3860

This theorem is referenced by:  cardcl  7210  isnumi  7211  cardval3ex  7214  lspval  13706  clsval  14068