Theorem elrabi 2837

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 2265	. . 3 |- ( A e. { x | ( x e. V /\ ph ) } <-> E. x ( x = A /\ ( x e. V /\ ph ) ) )
2		eleq1 2202	. . . . . 6 |- ( x = A -> ( x e. V <-> A e. V ) )
3	2	anbi1d 460	. . . . 5 |- ( x = A -> ( ( x e. V /\ ph ) <-> ( A e. V /\ ph ) ) )
4	3	simprbda 380	. . . 4 |- ( ( x = A /\ ( x e. V /\ ph ) ) -> A e. V )
5	4	exlimiv 1577	. . 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 2425	. 2 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
8	6, 7	eleq2s 2234	1 |- ( A e. { x e. V | ph } -> A e. V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2125   {crab 2420

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rab 2425

This theorem is referenced by:  ordtriexmidlem  4435  ordtri2or2exmidlem  4441  onsucelsucexmidlem  4444  ordsoexmid  4477  reg3exmidlemwe  4493  elfvmptrab1  5515  acexmidlemcase  5769  ssfirab  6822  exmidonfinlem  7049  genpelvl  7320  genpelvu  7321  suplocsrlempr  7615  nnindnn  7701  sup3exmid  8715  nnind  8736  supinfneg  9390  infsupneg  9391  supminfex  9392  ublbneg  9405  hashinfuni  10523  zsupcllemstep  11638  infssuzex  11642  infssuzledc  11643  bezoutlemsup  11697  lcmgcdlem  11758  oddennn  11905  evenennn  11906  znnen  11911  ennnfonelemg  11916  txdis1cn  12447  reopnap  12707  divcnap  12724  limccl  12797  dvlemap  12818  dvaddxxbr  12834  dvmulxxbr  12835  dvcoapbr  12840  dvcjbr  12841  dvrecap  12846  dveflem  12855