Theorem rspcedvd 2927

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2925. (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 2925	. 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 1398   e. wcel 2203   E.wrex 2521

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815

This theorem is referenced by:  rspcime  2928  rspcedeq1vd  2930  rspcedeq2vd  2931  updjud  7373  elpq  9981  modqmuladd  10728  modqmuladdnn0  10730  modfzo0difsn  10757  wrdl1exs1  11317  negfi  11913  divconjdvds  12535  2tp1odd  12570  dfgcd2  12710  qredeu  12794  pw2dvdslemn  12862  dvdsprmpweq  13033  oddprmdvds  13052  gsumfzval  13604  gsumval2  13610  isnsgrp  13619  dfgrp2  13740  grplrinv  13770  grpidinv  13772  dfgrp3m  13812  ringid  14170  xmettx  15375  gausslemma2dlem1a  15931  2lgslem1b  15962  usgredg4  16210  wlkvtxiedg  16340  wlkvtxiedgg  16341  umgr2cwwkdifex  16420  bj-charfunbi  16581