Theorem intexrabim 4201

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 4200	. 2 |- ( E. x ( x e. A /\ ph ) -> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2491	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2494	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 3891	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2272	. 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 1516   e. wcel 2177   {cab 2192   E.wrex 2486   {crab 2489   _Vcvv 2773   |^|cint 3887

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4166

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-in 3173  df-ss 3180  df-int 3888

This theorem is referenced by:  cardcl  7295  isnumi  7296  cardval3ex  7299  lspval  14196  clsval  14627