Theorem rspcedvd 2870

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2868. (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 2868	. 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 1364   e. wcel 2164   E.wrex 2473

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762

This theorem is referenced by:  rspcime  2871  rspcedeq1vd  2873  rspcedeq2vd  2874  updjud  7141  elpq  9714  modqmuladd  10437  modqmuladdnn0  10439  modfzo0difsn  10466  negfi  11371  divconjdvds  11991  2tp1odd  12025  dfgcd2  12151  qredeu  12235  pw2dvdslemn  12303  dvdsprmpweq  12473  oddprmdvds  12492  gsumfzval  12974  gsumval2  12980  isnsgrp  12989  dfgrp2  13099  grplrinv  13129  grpidinv  13131  dfgrp3m  13171  ringid  13522  xmettx  14678  gausslemma2dlem1a  15174  bj-charfunbi  15303