Theorem rspcedvd 2913

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2911. (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 2911	. 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 1395   e. wcel 2200   E.wrex 2509

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801

This theorem is referenced by:  rspcime  2914  rspcedeq1vd  2916  rspcedeq2vd  2917  updjud  7260  elpq  9856  modqmuladd  10600  modqmuladdnn0  10602  modfzo0difsn  10629  wrdl1exs1  11177  negfi  11754  divconjdvds  12375  2tp1odd  12410  dfgcd2  12550  qredeu  12634  pw2dvdslemn  12702  dvdsprmpweq  12873  oddprmdvds  12892  gsumfzval  13439  gsumval2  13445  isnsgrp  13454  dfgrp2  13575  grplrinv  13605  grpidinv  13607  dfgrp3m  13647  ringid  14004  xmettx  15199  gausslemma2dlem1a  15752  2lgslem1b  15783  usgredg4  16028  wlkvtxiedg  16086  wlkvtxiedgg  16087  bj-charfunbi  16229