Theorem rspcedvd 2935

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2933. (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 2933	. 2 |- ( ph -> ( ch -> E. x e. B ps ) )
5	1, 4	mpd 13	1 |- ( ph -> E. x e. B ps )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823

This theorem is used by:  rspcime  2937  rspcedeq1vd  2939  rspcedeq2vd  2940  updjud  7423  elpq  10060  modqmuladd  10818  modqmuladdnn0  10820  modfzo0difsn  10847  wrdl1exs1  11413  negfi  12011  divconjdvds  12635  2tp1odd  12670  dfgcd2  12810  qredeu  12894  dvdsprmpweq  13137  oddprmdvds  13156  isnsgrp  13774  dfgrp2  13885  grplrinv  13915  grpidinv  13917  dfgrp3m  13957  ringid  14415  xmettx  15702  gausslemma2dlem1a  16343  2lgslem1b  16374  usgredg4  16622  wlkvtxiedg  16752  wlkvtxiedgg  16753  umgr2cwwkdifex  16832  bj-charfunbi  17003