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Theorem rspcva 2710
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 2708 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32imp 122 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614
This theorem is referenced by:  supmoti  6595  peano2nnnn  7293  squeeze0  8259  peano2nn  8328  nnsub  8354  zextle  8733  rexuz3  10250  cau3lem  10374  caubnd2  10377  climcn1  10521  dvdsext  10636
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