Theorem elrabi 2891

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 2303	. . 3 |- ( A e. { x | ( x e. V /\ ph ) } <-> E. x ( x = A /\ ( x e. V /\ ph ) ) )
2		eleq1 2240	. . . . . 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 1598	. . 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 2464	. 2 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
8	6, 7	eleq2s 2272	1 |- ( A e. { x e. V | ph } -> A e. V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   {crab 2459

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-rab 2464

This theorem is referenced by:  ordtriexmidlem  4519  ordtri2or2exmidlem  4526  onsucelsucexmidlem  4529  ordsoexmid  4562  reg3exmidlemwe  4579  elfvmptrab1  5611  acexmidlemcase  5870  ssfirab  6933  exmidonfinlem  7192  cc4f  7268  genpelvl  7511  genpelvu  7512  suplocsrlempr  7806  nnindnn  7892  sup3exmid  8914  nnind  8935  supinfneg  9595  infsupneg  9596  supminfex  9597  ublbneg  9613  hashinfuni  10757  zsupcllemstep  11946  infssuzex  11950  infssuzledc  11951  bezoutlemsup  12010  uzwodc  12038  lcmgcdlem  12077  phisum  12240  oddennn  12393  evenennn  12394  znnen  12399  ennnfonelemg  12404  txdis1cn  13781  reopnap  14041  divcnap  14058  limccl  14131  dvlemap  14152  dvaddxxbr  14168  dvmulxxbr  14169  dvcoapbr  14174  dvcjbr  14175  dvrecap  14180  dveflem  14190