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Theorem rspcva 2792
 Description: Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
Hypothesis
Ref Expression
rspcv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcva ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rspcva
StepHypRef Expression
1 rspcv.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21rspcv 2790 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32imp 123 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ 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 2692 This theorem is referenced by:  supmoti  6890  peano2nnnn  7705  squeeze0  8706  peano2nn  8776  nnsub  8803  zextle  9186  rexuz3  10814  cau3lem  10938  caubnd2  10941  climcn1  11129  dvdsext  11609
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