Theorem rspcedvd 2871

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2869. (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 2869	. 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  2872  rspcedeq1vd  2874  rspcedeq2vd  2875  updjud  7143  elpq  9717  modqmuladd  10440  modqmuladdnn0  10442  modfzo0difsn  10469  negfi  11374  divconjdvds  11994  2tp1odd  12028  dfgcd2  12154  qredeu  12238  pw2dvdslemn  12306  dvdsprmpweq  12476  oddprmdvds  12495  gsumfzval  12977  gsumval2  12983  isnsgrp  12992  dfgrp2  13102  grplrinv  13132  grpidinv  13134  dfgrp3m  13174  ringid  13525  xmettx  14689  gausslemma2dlem1a  15215  2lgslem1b  15246  bj-charfunbi  15373