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Theorem bj-rspg 13008
Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2786 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 13007 . 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 1427 1  |-  ( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   F/wnf 1436    e. wcel 1480   F/_wnfc 2268   A.wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688
This theorem is referenced by:  bj-bdfindisg  13160  bj-findisg  13192
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