Theorem rspcedvd 2874

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv 2872. (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 2872	. 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 2167   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765

This theorem is referenced by:  rspcime  2875  rspcedeq1vd  2877  rspcedeq2vd  2878  updjud  7148  elpq  9723  modqmuladd  10458  modqmuladdnn0  10460  modfzo0difsn  10487  negfi  11393  divconjdvds  12014  2tp1odd  12049  dfgcd2  12181  qredeu  12265  pw2dvdslemn  12333  dvdsprmpweq  12504  oddprmdvds  12523  gsumfzval  13034  gsumval2  13040  isnsgrp  13049  dfgrp2  13159  grplrinv  13189  grpidinv  13191  dfgrp3m  13231  ringid  13582  xmettx  14746  gausslemma2dlem1a  15299  2lgslem1b  15330  bj-charfunbi  15457