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Theorem bj-rspg 9195
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 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 9194 . 2 (x(x = A → (φψ)) → (x B φ → (A Bψ)))
5 bj-rspg.is . 2 (x = A → (φψ))
64, 5mpg 1337 1 (x B φ → (A Bψ))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  wral 2300
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553
This theorem is referenced by:  bj-bdfindisg  9336  bj-findisg  9364
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