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  7245  elpq  9840  modqmuladd  10583  modqmuladdnn0  10585  modfzo0difsn  10612  wrdl1exs1  11157  negfi  11734  divconjdvds  12355  2tp1odd  12390  dfgcd2  12530  qredeu  12614  pw2dvdslemn  12682  dvdsprmpweq  12853  oddprmdvds  12872  gsumfzval  13419  gsumval2  13425  isnsgrp  13434  dfgrp2  13555  grplrinv  13585  grpidinv  13587  dfgrp3m  13627  ringid  13984  xmettx  15178  gausslemma2dlem1a  15731  2lgslem1b  15762  usgredg4  16007  wlkvtxiedgg  16042  bj-charfunbi  16132