ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rspcva GIF version

Theorem rspcva 2832
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 2830 . 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 1348  wcel 2141  wral 2448
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732
This theorem is referenced by:  supmoti  6970  peano2nnnn  7815  squeeze0  8820  peano2nn  8890  nnsub  8917  zextle  9303  rexuz3  10954  cau3lem  11078  caubnd2  11081  climcn1  11271  dvdsext  11815  mgmidmo  12626
  Copyright terms: Public domain W3C validator