Theorem elrabd 2931

Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2929. (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 2929	. 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 1373   e. wcel 2176   {crab 2488

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774

This theorem is referenced by:  ctssdccl  7215  suplocexprlemru  7834  suplocexprlemloc  7836  zsupssdc  10383  uzwodc  12391  nninfctlemfo  12394  lcmcllem  12422  lcmledvds  12425  phisum  12596  odzcllem  12598  pcpremul  12649  znnen  12802  ennnfonelemj0  12805  ennnfonelemg  12807  gsumress  13260  issubmd  13339  mhmeql  13357  ghmeql  13636  cdivcncfap  15109  cnopnap  15116  ivthinc  15148  limcdifap  15167  limcimolemlt  15169  dvcoapbr  15212  dvdsppwf1o  15494  2lgslem1b  15599  2omap  15969  subctctexmid  15974