Theorem intexrabim 3907

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 3906	. 2 |- ( E. x ( x e. A /\ ph ) -> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2312	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2315	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 3619	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2103	. 2 |- ( |^| { x e. A | ph } e. _V <-> |^| { x | ( x e. A /\ ph ) } e. _V )
6	1, 2, 5	3imtr4i 190	1 |- ( E. x e. A ph -> |^| { x e. A | ph } e. _V )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   {crab 2310   _Vcvv 2557   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616

This theorem is referenced by:  cardcl  6361  isnumi  6362  cardval3ex  6365