Theorem rspcedvd 2914

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2912. (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 2912	. 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 2802

This theorem is referenced by:  rspcime  2915  rspcedeq1vd  2917  rspcedeq2vd  2918  updjud  7272  elpq  9873  modqmuladd  10618  modqmuladdnn0  10620  modfzo0difsn  10647  wrdl1exs1  11196  negfi  11779  divconjdvds  12400  2tp1odd  12435  dfgcd2  12575  qredeu  12659  pw2dvdslemn  12727  dvdsprmpweq  12898  oddprmdvds  12917  gsumfzval  13464  gsumval2  13470  isnsgrp  13479  dfgrp2  13600  grplrinv  13630  grpidinv  13632  dfgrp3m  13672  ringid  14029  xmettx  15224  gausslemma2dlem1a  15777  2lgslem1b  15808  usgredg4  16054  wlkvtxiedg  16142  wlkvtxiedgg  16143  umgr2cwwkdifex  16220  bj-charfunbi  16342