Theorem rspcedvd 2883

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2881. (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 2881	. 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 1373   e. wcel 2176   E.wrex 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774

This theorem is referenced by:  rspcime  2884  rspcedeq1vd  2886  rspcedeq2vd  2887  updjud  7184  elpq  9770  modqmuladd  10511  modqmuladdnn0  10513  modfzo0difsn  10540  wrdl1exs1  11083  negfi  11539  divconjdvds  12160  2tp1odd  12195  dfgcd2  12335  qredeu  12419  pw2dvdslemn  12487  dvdsprmpweq  12658  oddprmdvds  12677  gsumfzval  13223  gsumval2  13229  isnsgrp  13238  dfgrp2  13359  grplrinv  13389  grpidinv  13391  dfgrp3m  13431  ringid  13788  xmettx  14982  gausslemma2dlem1a  15535  2lgslem1b  15566  bj-charfunbi  15747