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

Theorem rspccva 2788
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
rspcv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspccva ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rspccva
StepHypRef Expression
1 rspcv.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21rspcv 2785 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32impcom 124 1 ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688
This theorem is referenced by:  disjne  3416  seex  4257  fconstfvm  5638  grprinvlem  5965  fvixp  6597  ordiso2  6920  eqord1  8252  eqord2  8253  seq3caopr2  10262  bccl  10520  2clim  11077  isummulc2  11202  telfsumo2  11243  fsumparts  11246  isumshft  11266  mertenslem2  11312  mertensabs  11313  dvdsprime  11809  cnima  12398
  Copyright terms: Public domain W3C validator