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Theorem rspccv 2838
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 2837 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32com12 30 1 (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  wral 2455
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739
This theorem is referenced by:  elinti  3851  ofrval  6086  supubti  6991  suplubti  6992  suplocsrlempr  7784  pitonn  7825  peano5uzti  9337  zindd  9347  1arith  12335  basis2  13179  tg2  13193  mopni  13615  metrest  13639  metcnpi  13648  metcnpi2  13649  decidi  14169  sumdc2  14173
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