Theorem elrabd 2922

Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2920. (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 2920	. 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 1364   e. wcel 2167   {crab 2479

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765

This theorem is referenced by:  ctssdccl  7177  suplocexprlemru  7786  suplocexprlemloc  7788  zsupssdc  10328  uzwodc  12204  nninfctlemfo  12207  lcmcllem  12235  lcmledvds  12238  phisum  12409  odzcllem  12411  pcpremul  12462  znnen  12615  ennnfonelemj0  12618  ennnfonelemg  12620  gsumress  13038  issubmd  13106  mhmeql  13124  ghmeql  13397  cdivcncfap  14840  cnopnap  14847  ivthinc  14879  limcdifap  14898  limcimolemlt  14900  dvcoapbr  14943  dvdsppwf1o  15225  2lgslem1b  15330  subctctexmid  15645