Theorem elrabi 2917

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 2322	. . 3 |- ( A e. { x | ( x e. V /\ ph ) } <-> E. x ( x = A /\ ( x e. V /\ ph ) ) )
2		eleq1 2259	. . . . . 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 1612	. . 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 2484	. 2 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
8	6, 7	eleq2s 2291	1 |- ( A e. { x e. V | ph } -> A e. V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   {crab 2479

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rab 2484

This theorem is referenced by:  ordtriexmidlem  4556  ordtri2or2exmidlem  4563  onsucelsucexmidlem  4566  ordsoexmid  4599  reg3exmidlemwe  4616  elfvmptrab1  5659  acexmidlemcase  5920  elovmporab  6127  elovmporab1w  6128  ssfirab  7006  exmidonfinlem  7272  cc4f  7352  genpelvl  7596  genpelvu  7597  suplocsrlempr  7891  nnindnn  7977  sup3exmid  9001  nnind  9023  supinfneg  9686  infsupneg  9687  supminfex  9688  ublbneg  9704  zsupcllemstep  10336  infssuzex  10340  infssuzledc  10341  hashinfuni  10886  bezoutlemsup  12201  uzwodc  12229  nninfctlemfo  12232  lcmgcdlem  12270  phisum  12434  oddennn  12634  evenennn  12635  znnen  12640  ennnfonelemg  12645  rrgval  13894  psrbagf  14300  txdis1cn  14598  reopnap  14866  divcnap  14885  limccl  14979  dvlemap  15000  dvaddxxbr  15021  dvmulxxbr  15022  dvcoapbr  15027  dvcjbr  15028  dvrecap  15033  dveflem  15046  sgmval  15303  0sgm  15305  sgmf  15306  sgmnncl  15308  dvdsppwf1o  15309  sgmppw  15312