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Theorem bj-rspg 10733
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2699 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-rspg.nfa  |-  F/_ x A
bj-rspg.nfb  |-  F/_ x B
bj-rspg.nf2  |-  F/ x ps
bj-rspg.is  |-  ( x  =  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
bj-rspg  |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) )

Proof of Theorem bj-rspg
StepHypRef Expression
1 bj-rspg.nfa . . 3  |-  F/_ x A
2 bj-rspg.nfb . . 3  |-  F/_ x B
3 bj-rspg.nf2 . . 3  |-  F/ x ps
41, 2, 3bj-rspgt 10732 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) ) )
5 bj-rspg.is . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
64, 5mpg 1381 1  |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   F/wnf 1390    e. wcel 1434   F/_wnfc 2207   A.wral 2349
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604
This theorem is referenced by:  bj-bdfindisg  10886  bj-findisg  10918
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