Theorem rspcedvd 2890

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2888. (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 2888	. 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 2178   E.wrex 2487

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778

This theorem is referenced by:  rspcime  2891  rspcedeq1vd  2893  rspcedeq2vd  2894  updjud  7210  elpq  9805  modqmuladd  10548  modqmuladdnn0  10550  modfzo0difsn  10577  wrdl1exs1  11121  negfi  11654  divconjdvds  12275  2tp1odd  12310  dfgcd2  12450  qredeu  12534  pw2dvdslemn  12602  dvdsprmpweq  12773  oddprmdvds  12792  gsumfzval  13338  gsumval2  13344  isnsgrp  13353  dfgrp2  13474  grplrinv  13504  grpidinv  13506  dfgrp3m  13546  ringid  13903  xmettx  15097  gausslemma2dlem1a  15650  2lgslem1b  15681  bj-charfunbi  15946