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  7274  cc4f  7354  genpelvl  7598  genpelvu  7599  suplocsrlempr  7893  nnindnn  7979  sup3exmid  9003  nnind  9025  supinfneg  9688  infsupneg  9689  supminfex  9690  ublbneg  9706  zsupcllemstep  10338  infssuzex  10342  infssuzledc  10343  hashinfuni  10888  bezoutlemsup  12203  uzwodc  12231  nninfctlemfo  12234  lcmgcdlem  12272  phisum  12436  oddennn  12636  evenennn  12637  znnen  12642  ennnfonelemg  12647  rrgval  13896  psrbagf  14302  txdis1cn  14600  reopnap  14868  divcnap  14887  limccl  14981  dvlemap  15002  dvaddxxbr  15023  dvmulxxbr  15024  dvcoapbr  15029  dvcjbr  15030  dvrecap  15035  dveflem  15048  sgmval  15305  0sgm  15307  sgmf  15308  sgmnncl  15310  dvdsppwf1o  15311  sgmppw  15314