Theorem rspcedvd 2935

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2933. (Contributed by AV, 27-Nov-2019.)

Hypotheses

Ref	Expression
rspcedvd.1	|- ( ph -> A e. B )
rspcedvd.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
rspcedvd.3	|- ( ph -> ch )

Assertion

Ref	Expression
rspcedvd	|- ( ph -> E. x e. B ps )

Distinct variable groups:   x, A   x, B   ph, x   ch, x

Allowed substitution hint:   ps( x)

Proof of Theorem rspcedvd

Step	Hyp	Ref	Expression
1		rspcedvd.3	. 2 |- ( ph -> ch )
2		rspcedvd.1	. . 3 |- ( ph -> A e. B )
3		rspcedvd.2	. . 3 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4	2, 3	rspcedv 2933	. 2 |- ( ph -> ( ch -> E. x e. B ps ) )
5	1, 4	mpd 13	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823

This theorem is referenced by:  rspcime  2937  rspcedeq1vd  2939  rspcedeq2vd  2940  updjud  7412  elpq  10028  modqmuladd  10781  modqmuladdnn0  10783  modfzo0difsn  10810  wrdl1exs1  11375  negfi  11972  divconjdvds  12594  2tp1odd  12629  dfgcd2  12769  qredeu  12853  pw2dvdslemn  12921  dvdsprmpweq  13092  oddprmdvds  13111  isnsgrp  13698  dfgrp2  13809  grplrinv  13839  grpidinv  13841  dfgrp3m  13881  ringid  14304  xmettx  15534  gausslemma2dlem1a  16091  2lgslem1b  16122  usgredg4  16370  wlkvtxiedg  16500  wlkvtxiedgg  16501  umgr2cwwkdifex  16580  bj-charfunbi  16751