Theorem elrabd 2961

Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2959. (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 306	. 2 |- ( ph -> ( A e. B /\ ch ) )
4		elrabd.1	. . 3 |- ( x = A -> ( ps <-> ch ) )
5	4	elrab 2959	. 2 |- ( A e. { x e. B | ps } <-> ( A e. B /\ ch ) )
6	3, 5	sylibr 134	1 |- ( ph -> A e. { x e. B | ps } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801

This theorem is referenced by:  ctssdccl  7278  suplocexprlemru  7906  suplocexprlemloc  7908  zsupssdc  10458  uzwodc  12558  nninfctlemfo  12561  lcmcllem  12589  lcmledvds  12592  phisum  12763  odzcllem  12765  pcpremul  12816  znnen  12969  ennnfonelemj0  12972  ennnfonelemg  12974  gsumress  13428  issubmd  13507  mhmeql  13525  ghmeql  13804  cdivcncfap  15278  cnopnap  15285  ivthinc  15317  limcdifap  15336  limcimolemlt  15338  dvcoapbr  15381  dvdsppwf1o  15663  2lgslem1b  15768  incistruhgr  15890  upgr1elem1  15920  2omap  16359  subctctexmid  16366