Theorem rspcedvd 2916

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2914. (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 2914	. 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 1397   e. wcel 2202   E.wrex 2511

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804

This theorem is referenced by:  rspcime  2917  rspcedeq1vd  2919  rspcedeq2vd  2920  updjud  7281  elpq  9883  modqmuladd  10629  modqmuladdnn0  10631  modfzo0difsn  10658  wrdl1exs1  11207  negfi  11790  divconjdvds  12412  2tp1odd  12447  dfgcd2  12587  qredeu  12671  pw2dvdslemn  12739  dvdsprmpweq  12910  oddprmdvds  12929  gsumfzval  13476  gsumval2  13482  isnsgrp  13491  dfgrp2  13612  grplrinv  13642  grpidinv  13644  dfgrp3m  13684  ringid  14042  xmettx  15237  gausslemma2dlem1a  15790  2lgslem1b  15821  usgredg4  16069  wlkvtxiedg  16199  wlkvtxiedgg  16200  umgr2cwwkdifex  16279  bj-charfunbi  16423