Theorem elrabi 2879

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 2292	. . 3 |- ( A e. { x | ( x e. V /\ ph ) } <-> E. x ( x = A /\ ( x e. V /\ ph ) ) )
2		eleq1 2229	. . . . . 6 |- ( x = A -> ( x e. V <-> A e. V ) )
3	2	anbi1d 461	. . . . 5 |- ( x = A -> ( ( x e. V /\ ph ) <-> ( A e. V /\ ph ) ) )
4	3	simprbda 381	. . . 4 |- ( ( x = A /\ ( x e. V /\ ph ) ) -> A e. V )
5	4	exlimiv 1586	. . 3 |- ( E. x ( x = A /\ ( x e. V /\ ph ) ) -> A e. V )
6	1, 5	sylbi 120	. 2 |- ( A e. { x | ( x e. V /\ ph ) } -> A e. V )
7		df-rab 2453	. 2 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
8	6, 7	eleq2s 2261	1 |- ( A e. { x e. V | ph } -> A e. V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-rab 2453

This theorem is referenced by:  ordtriexmidlem  4496  ordtri2or2exmidlem  4503  onsucelsucexmidlem  4506  ordsoexmid  4539  reg3exmidlemwe  4556  elfvmptrab1  5580  acexmidlemcase  5837  ssfirab  6899  exmidonfinlem  7149  cc4f  7210  genpelvl  7453  genpelvu  7454  suplocsrlempr  7748  nnindnn  7834  sup3exmid  8852  nnind  8873  supinfneg  9533  infsupneg  9534  supminfex  9535  ublbneg  9551  hashinfuni  10690  zsupcllemstep  11878  infssuzex  11882  infssuzledc  11883  bezoutlemsup  11942  uzwodc  11970  lcmgcdlem  12009  phisum  12172  oddennn  12325  evenennn  12326  znnen  12331  ennnfonelemg  12336  txdis1cn  12918  reopnap  13178  divcnap  13195  limccl  13268  dvlemap  13289  dvaddxxbr  13305  dvmulxxbr  13306  dvcoapbr  13311  dvcjbr  13312  dvrecap  13317  dveflem  13327