Theorem elrabi 2913

Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Assertion

Ref	Expression
elrabi	|- ( A e. { x e. V | ph } -> A e. V )

Distinct variable groups:   x, A   x, V

Allowed substitution hint:   ph( x)

Proof of Theorem elrabi

Step	Hyp	Ref	Expression
1		clelab 2319	. . 3 |- ( A e. { x | ( x e. V /\ ph ) } <-> E. x ( x = A /\ ( x e. V /\ ph ) ) )
2		eleq1 2256	. . . . . 6 |- ( x = A -> ( x e. V <-> A e. V ) )
3	2	anbi1d 465	. . . . 5 |- ( x = A -> ( ( x e. V /\ ph ) <-> ( A e. V /\ ph ) ) )
4	3	simprbda 383	. . . 4 |- ( ( x = A /\ ( x e. V /\ ph ) ) -> A e. V )
5	4	exlimiv 1609	. . 3 |- ( E. x ( x = A /\ ( x e. V /\ ph ) ) -> A e. V )
6	1, 5	sylbi 121	. 2 |- ( A e. { x | ( x e. V /\ ph ) } -> A e. V )
7		df-rab 2481	. 2 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
8	6, 7	eleq2s 2288	1 |- ( A e. { x e. V | ph } -> A e. V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   {crab 2476

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rab 2481

This theorem is referenced by:  ordtriexmidlem  4551  ordtri2or2exmidlem  4558  onsucelsucexmidlem  4561  ordsoexmid  4594  reg3exmidlemwe  4611  elfvmptrab1  5652  acexmidlemcase  5913  elovmporab  6118  elovmporab1w  6119  ssfirab  6990  exmidonfinlem  7253  cc4f  7329  genpelvl  7572  genpelvu  7573  suplocsrlempr  7867  nnindnn  7953  sup3exmid  8976  nnind  8998  supinfneg  9660  infsupneg  9661  supminfex  9662  ublbneg  9678  hashinfuni  10848  zsupcllemstep  12082  infssuzex  12086  infssuzledc  12087  bezoutlemsup  12146  uzwodc  12174  nninfctlemfo  12177  lcmgcdlem  12215  phisum  12378  oddennn  12549  evenennn  12550  znnen  12555  ennnfonelemg  12560  rrgval  13758  psrbagf  14156  txdis1cn  14446  reopnap  14706  divcnap  14723  limccl  14813  dvlemap  14834  dvaddxxbr  14850  dvmulxxbr  14851  dvcoapbr  14856  dvcjbr  14857  dvrecap  14862  dveflem  14872