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Theorem rspcva 2671
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 2669 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32imp 119 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by:  tfisi  4337  suppssov1  5736  caofinvl  5760  tfrlem1  5953  supmoti  6398  caucvgsrlemgt1  6936  peano2nnnn  6986  axcaucvglemcau  7029  squeeze0  7944  peano2nn  8001  nnsub  8027  zextle  8388  rexuz3  9816  cau3lem  9940  caubnd2  9943  climcn1  10059  serif0  10101  dvdsext  10166
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