Theorem rspcedvd 2929

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2927. (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 2927	. 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 2205   E.wrex 2523

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817

This theorem is referenced by:  rspcime  2931  rspcedeq1vd  2933  rspcedeq2vd  2934  updjud  7386  elpq  9999  modqmuladd  10752  modqmuladdnn0  10754  modfzo0difsn  10781  wrdl1exs1  11342  negfi  11938  divconjdvds  12560  2tp1odd  12595  dfgcd2  12735  qredeu  12819  pw2dvdslemn  12887  dvdsprmpweq  13058  oddprmdvds  13077  gsumfzval  13654  gsumval2  13660  isnsgrp  13669  dfgrp2  13782  grplrinv  13812  grpidinv  13814  dfgrp3m  13854  ringid  14269  xmettx  15501  gausslemma2dlem1a  16057  2lgslem1b  16088  usgredg4  16336  wlkvtxiedg  16466  wlkvtxiedgg  16467  umgr2cwwkdifex  16546  bj-charfunbi  16707