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Theorem rspccva 2841
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 2838 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32impcom 125 1 ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740
This theorem is referenced by:  disjne  3477  seex  4336  fconstfvm  5735  fvixp  6703  ordiso2  7034  eqord1  8440  eqord2  8441  seq3caopr2  10482  bccl  10747  2clim  11309  isummulc2  11434  telfsumo2  11475  fsumparts  11478  isumshft  11498  mertenslem2  11544  mertensabs  11545  dvdsprime  12122  mgmlrid  12798  grprinvlem  12804  grpinvex  12887  issubg2m  13049  issubg4m  13053  nmzbi  13069  cnima  13723  dceqnconst  14810  dcapnconst  14811
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