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Theorem spcegv 2635
Description: Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
spcgv.1 (x = A → (φψ))
Assertion
Ref Expression
spcegv (A 𝑉 → (ψxφ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem spcegv
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfv 1418 . 2 xψ
3 spcgv.1 . 2 (x = A → (φψ))
41, 2, 3spcegf 2630 1 (A 𝑉 → (ψxφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  spcev  2641  eqeu  2705  absneu  3433  elunii  3576  axpweq  3915  euotd  3982  brcogw  4447  opeldmg  4483  breldmg  4484  dmsnopg  4735  dff3im  5255  elunirn  5348  unielxp  5742  op1steq  5747  tfr0  5878  tfrlemibxssdm  5882  tfrlemiex  5886  ertr  6057  f1oen3g  6170  f1dom2g  6172  f1domg  6174  dom3d  6190  en1  6215  recexnq  6374  ltexprlemrl  6583  ltexprlemru  6585  recexprlemm  6595  recexprlemloc  6602  recexprlem1ssl  6604  recexprlem1ssu  6605  frecuzrdgfn  8859  bj-2inf  9372
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