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  7422  elpq  10059  modqmuladd  10816  modqmuladdnn0  10818  modfzo0difsn  10845  wrdl1exs1  11411  negfi  12009  divconjdvds  12632  2tp1odd  12667  dfgcd2  12807  qredeu  12891  dvdsprmpweq  13134  oddprmdvds  13153  isnsgrp  13770  dfgrp2  13881  grplrinv  13911  grpidinv  13913  dfgrp3m  13953  ringid  14380  xmettx  15660  gausslemma2dlem1a  16275  2lgslem1b  16306  usgredg4  16554  wlkvtxiedg  16684  wlkvtxiedgg  16685  umgr2cwwkdifex  16764  bj-charfunbi  16935