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  10049  modqmuladd  10803  modqmuladdnn0  10805  modfzo0difsn  10832  wrdl1exs1  11397  negfi  11994  divconjdvds  12616  2tp1odd  12651  dfgcd2  12791  qredeu  12875  pw2dvdslemn  12943  dvdsprmpweq  13114  oddprmdvds  13133  isnsgrp  13721  dfgrp2  13832  grplrinv  13862  grpidinv  13864  dfgrp3m  13904  ringid  14331  xmettx  15611  gausslemma2dlem1a  16177  2lgslem1b  16208  usgredg4  16456  wlkvtxiedg  16586  wlkvtxiedgg  16587  umgr2cwwkdifex  16666  bj-charfunbi  16837