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Theorem rspccv 2789
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
Hypothesis
Ref Expression
rspcv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspccv (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rspccv
StepHypRef Expression
1 rspcv.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21rspcv 2788 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32com12 30 1 (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wcel 1481  wral 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691
This theorem is referenced by:  elinti  3787  ofrval  5999  supubti  6893  suplubti  6894  suplocsrlempr  7638  pitonn  7679  peano5uzti  9182  zindd  9192  basis2  12252  tg2  12266  mopni  12688  metrest  12712  metcnpi  12721  metcnpi2  12722  decidi  13171  sumdc2  13175
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