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Theorem bj-rspg 11991
 Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2722 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-rspg.nfa 𝑥𝐴
bj-rspg.nfb 𝑥𝐵
bj-rspg.nf2 𝑥𝜓
bj-rspg.is (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
bj-rspg (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))

Proof of Theorem bj-rspg
StepHypRef Expression
1 bj-rspg.nfa . . 3 𝑥𝐴
2 bj-rspg.nfb . . 3 𝑥𝐵
3 bj-rspg.nf2 . . 3 𝑥𝜓
41, 2, 3bj-rspgt 11990 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
5 bj-rspg.is . 2 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5mpg 1386 1 (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290  Ⅎwnf 1395   ∈ wcel 1439  Ⅎwnfc 2216  ∀wral 2360 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624 This theorem is referenced by:  bj-bdfindisg  12147  bj-findisg  12179
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