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Theorem rspcva 2862
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 2860 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
32imp 124 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  wral 2472
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762
This theorem is referenced by:  supmoti  7052  peano2nnnn  7913  squeeze0  8923  peano2nn  8994  nnsub  9021  zextle  9408  rexuz3  11134  cau3lem  11258  caubnd2  11261  climcn1  11451  dvdsext  11997  mgmidmo  12955  dfgrp3mlem  13170
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