Theorem intexrabim 4148

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 4147	. 2 |- ( E. x ( x e. A /\ ph ) -> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2459	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2462	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 3844	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2241	. 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 1490   e. wcel 2146   {cab 2161   E.wrex 2454   {crab 2457   _Vcvv 2735   |^|cint 3840

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-in 3133  df-ss 3140  df-int 3841

This theorem is referenced by:  cardcl  7170  isnumi  7171  cardval3ex  7174  clsval  13182