Theorem elrabd 3365

Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3363. (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 554	. 2 |- ( ph -> ( A e. B /\ ch ) )
4		elrabd.1	. . 3 |- ( x = A -> ( ps <-> ch ) )
5	4	elrab 3363	. 2 |- ( A e. { x e. B | ps } <-> ( A e. B /\ ch ) )
6	3, 5	sylibr 224	1 |- ( ph -> A e. { x e. B | ps } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202

This theorem is referenced by:  lcmgcdlem  15319  upgrreslem  26196  umgrreslem  26197  supminfrnmpt  39672  supminfxr  39694  supminfxr2  39699  supminfxrrnmpt  39701  smflimsuplem5  41030