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Theorem bj-rspg 10857
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2707 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 10856 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
5 bj-rspg.is . 2 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5mpg 1381 1 (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wnf 1390  wcel 1434  wnfc 2210  wral 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612
This theorem is referenced by:  bj-bdfindisg  11010  bj-findisg  11042
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