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  7289  suplocexprlemru  7917  suplocexprlemloc  7919  zsupssdc  10470  uzwodc  12574  nninfctlemfo  12577  lcmcllem  12605  lcmledvds  12608  phisum  12779  odzcllem  12781  pcpremul  12832  znnen  12985  ennnfonelemj0  12988  ennnfonelemg  12990  gsumress  13444  issubmd  13523  mhmeql  13541  ghmeql  13820  cdivcncfap  15294  cnopnap  15301  ivthinc  15333  limcdifap  15352  limcimolemlt  15354  dvcoapbr  15397  dvdsppwf1o  15679  2lgslem1b  15784  incistruhgr  15906  upgr1elem1  15936  2omap  16446  subctctexmid  16453