Theorem elrabd 2870

Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2868. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
elrabd.1	|- ( x = A -> ( ps <-> ch ) )
elrabd.2	|- ( ph -> A e. B )
elrabd.3	|- ( ph -> ch )

Assertion

Ref	Expression
elrabd	|- ( ph -> A e. { x e. B | ps } )

Distinct variable groups:   x, A   x, B   ch, x

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem elrabd

Step	Hyp	Ref	Expression
1		elrabd.2	. . 3 |- ( ph -> A e. B )
2		elrabd.3	. . 3 |- ( ph -> ch )
3	1, 2	jca 304	. 2 |- ( ph -> ( A e. B /\ ch ) )
4		elrabd.1	. . 3 |- ( x = A -> ( ps <-> ch ) )
5	4	elrab 2868	. 2 |- ( A e. { x e. B | ps } <-> ( A e. B /\ ch ) )
6	3, 5	sylibr 133	1 |- ( ph -> A e. { x e. B | ps } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {crab 2439

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714

This theorem is referenced by:  ctssdccl  7050  suplocexprlemru  7634  suplocexprlemloc  7636  phisum  12107  znnen  12114  ennnfonelemj0  12117  ennnfonelemg  12119  cdivcncfap  12974  cnopnap  12981  ivthinc  13008  limcdifap  13018  limcimolemlt  13020  dvcoapbr  13058  subctctexmid  13560