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  11210  negfi  11806  divconjdvds  12428  2tp1odd  12463  dfgcd2  12603  qredeu  12687  pw2dvdslemn  12755  dvdsprmpweq  12926  oddprmdvds  12945  gsumfzval  13492  gsumval2  13498  isnsgrp  13507  dfgrp2  13628  grplrinv  13658  grpidinv  13660  dfgrp3m  13700  ringid  14058  xmettx  15253  gausslemma2dlem1a  15806  2lgslem1b  15837  usgredg4  16085  wlkvtxiedg  16215  wlkvtxiedgg  16216  umgr2cwwkdifex  16295  bj-charfunbi  16457