Theorem elrabi 2914

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  4552  ordtri2or2exmidlem  4559  onsucelsucexmidlem  4562  ordsoexmid  4595  reg3exmidlemwe  4612  elfvmptrab1  5653  acexmidlemcase  5914  elovmporab  6120  elovmporab1w  6121  ssfirab  6992  exmidonfinlem  7255  cc4f  7331  genpelvl  7574  genpelvu  7575  suplocsrlempr  7869  nnindnn  7955  sup3exmid  8978  nnind  9000  supinfneg  9663  infsupneg  9664  supminfex  9665  ublbneg  9681  hashinfuni  10851  zsupcllemstep  12085  infssuzex  12089  infssuzledc  12090  bezoutlemsup  12149  uzwodc  12177  nninfctlemfo  12180  lcmgcdlem  12218  phisum  12381  oddennn  12552  evenennn  12553  znnen  12558  ennnfonelemg  12563  rrgval  13761  psrbagf  14167  txdis1cn  14457  reopnap  14725  divcnap  14744  limccl  14838  dvlemap  14859  dvaddxxbr  14880  dvmulxxbr  14881  dvcoapbr  14886  dvcjbr  14887  dvrecap  14892  dveflem  14905