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Theorem rspcev 2629
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
Hypothesis
Ref Expression
rspcv.1 (x = A → (φψ))
Assertion
Ref Expression
rspcev ((A B ψ) → x B φ)
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem rspcev
StepHypRef Expression
1 nfv 1398 . 2 xψ
2 rspcv.1 . 2 (x = A → (φψ))
31, 2rspce 2624 1 ((A B ψ) → x B φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  wrex 2281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533
This theorem is referenced by:  rspc2ev  2637  rspc3ev  2639  reu6i  2705  rspesbca  2815  nn0suc  4250  elrnmpt1s  4507  elrnrexdm  5227  eldmrexrn  5229  foco2  5239  elabrex  5318  f1elima  5333  fcofo  5345  fliftfun  5357  fliftval  5361  f1oiso2  5387  fo1st  5703  fo2nd  5704  tfrlemisucaccv  5856  tfrlemi14d  5864  tfrlemi14  5865  tfrexlem  5866  rdgss  5886  nnaordex  6007  nnawordex  6008  ecelqsg  6066  archnqq  6268  prarloclemarch2  6270  prcdnql  6332  prcunqu  6333  prarloclemlo  6342  prarloclem5  6348  nqprm  6391  1idprl  6423  1idpru  6424  ltexpri  6444  recexprlemm  6452  recexprlem1ssl  6461  recexprlem1ssu  6462  recexpr  6466  negexsr  6513  recexsrlem  6514  axrnegex  6567  axprecex  6568  cnegex  6776  bj-nn0suc0  7311  bj-inf2vnlem1  7327
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