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Theorem spcev 2641
Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
Hypotheses
Ref Expression
spcv.1 A V
spcv.2 (x = A → (φψ))
Assertion
Ref Expression
spcev (ψxφ)
Distinct variable groups:   x,A   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem spcev
StepHypRef Expression
1 spcv.1 . 2 A V
2 spcv.2 . . 3 (x = A → (φψ))
32spcegv 2635 . 2 (A V → (ψxφ))
41, 3ax-mp 7 1 (ψxφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551
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:  bnd2  3917  mss  3953  exss  3954  snnex  4147  opeldm  4481  elrnmpt1  4528  xpmlem  4687  ffoss  5101  ssimaex  5177  fvelrn  5241  eufnfv  5332  foeqcnvco  5373  domtr  6201  ensn1  6212
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