Theorem rspcedvd 2917

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2915. (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 2915	. 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 2202   E.wrex 2512

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805

This theorem is referenced by:  rspcime  2918  rspcedeq1vd  2920  rspcedeq2vd  2921  updjud  7324  elpq  9927  modqmuladd  10674  modqmuladdnn0  10676  modfzo0difsn  10703  wrdl1exs1  11255  negfi  11851  divconjdvds  12473  2tp1odd  12508  dfgcd2  12648  qredeu  12732  pw2dvdslemn  12800  dvdsprmpweq  12971  oddprmdvds  12990  gsumfzval  13537  gsumval2  13543  isnsgrp  13552  dfgrp2  13673  grplrinv  13703  grpidinv  13705  dfgrp3m  13745  ringid  14103  xmettx  15304  gausslemma2dlem1a  15860  2lgslem1b  15891  usgredg4  16139  wlkvtxiedg  16269  wlkvtxiedgg  16270  umgr2cwwkdifex  16349  bj-charfunbi  16510