Theorem intexrabim 4078

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 4077	. 2 |- ( E. x ( x e. A /\ ph ) -> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2422	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2425	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 3775	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2205	. 2 |- ( |^| { x e. A | ph } e. _V <-> |^| { x | ( x e. A /\ ph ) } e. _V )
6	1, 2, 5	3imtr4i 200	1 |- ( E. x e. A ph -> |^| { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1468   e. wcel 1480   {cab 2125   E.wrex 2417   {crab 2420   _Vcvv 2686   |^|cint 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-int 3772

This theorem is referenced by:  cardcl  7037  isnumi  7038  cardval3ex  7041  clsval  12280