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Theorem rspccva 2842
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 2839 . 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 2741
This theorem is referenced by:  disjne  3478  seex  4337  fconstfvm  5736  fvixp  6705  ordiso2  7036  eqord1  8442  eqord2  8443  seq3caopr2  10484  bccl  10749  2clim  11311  isummulc2  11436  telfsumo2  11477  fsumparts  11480  isumshft  11500  mertenslem2  11546  mertensabs  11547  dvdsprime  12124  mgmlrid  12803  grprinvlem  12809  grpinvex  12892  issubg2m  13054  issubg4m  13058  nmzbi  13074  cnima  13759  dceqnconst  14847  dcapnconst  14848
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