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Theorem bj-rspg 7033
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2630 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 (x B φ → (A Bψ))

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 7032 . 2 (x(x = A → (φψ)) → (x B φ → (A Bψ)))
5 bj-rspg.is . 2 (x = A → (φψ))
64, 5mpg 1320 1 (x B φ → (A Bψ))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wnf 1329   wcel 1374  wnfc 2147  wral 2284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537
This theorem is referenced by:  bj-bdfindisg  7170  bj-findisg  7198
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