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Theorem bj-rspgt 10747
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
Assertion
Ref Expression
bj-rspgt  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) ) )

Proof of Theorem bj-rspgt
StepHypRef Expression
1 eleq1 2142 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
21imbi1d 229 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  e.  B  ->  ( A. x  e.  B  ph  ->  ph )
)  <->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ph ) ) ) )
32biimpd 142 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  B  ->  ( A. x  e.  B  ph  ->  ph )
)  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ph ) ) ) )
4 imim2 54 . . . . . . . 8  |-  ( (
ph  ->  ps )  -> 
( ( A. x  e.  B  ph  ->  ph )  ->  ( A. x  e.  B  ph  ->  ps ) ) )
54imim2d 53 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ph ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) ) ) )
63, 5syl9 71 . . . . . 6  |-  ( x  =  A  ->  (
( ph  ->  ps )  ->  ( ( x  e.  B  ->  ( A. x  e.  B  ph  ->  ph ) )  ->  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  ps ) ) ) ) )
76a2i 11 . . . . 5  |-  ( ( x  =  A  -> 
( ph  ->  ps )
)  ->  ( x  =  A  ->  ( ( x  e.  B  -> 
( A. x  e.  B  ph  ->  ph )
)  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) ) ) )
87alimi 1385 . . . 4  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  ->  A. x ( x  =  A  ->  ( (
x  e.  B  -> 
( A. x  e.  B  ph  ->  ph )
)  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) ) ) )
9 bj-rspg.nfa . . . . 5  |-  F/_ x A
10 bj-rspg.nfb . . . . . . 7  |-  F/_ x B
119, 10nfel 2228 . . . . . 6  |-  F/ x  A  e.  B
12 nfra1 2398 . . . . . . 7  |-  F/ x A. x  e.  B  ph
13 bj-rspg.nf2 . . . . . . 7  |-  F/ x ps
1412, 13nfim 1505 . . . . . 6  |-  F/ x
( A. x  e.  B  ph  ->  ps )
1511, 14nfim 1505 . . . . 5  |-  F/ x
( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) )
16 rsp 2412 . . . . . . 7  |-  ( A. x  e.  B  ph  ->  ( x  e.  B  ->  ph ) )
1716a1i 9 . . . . . 6  |-  ( x  =  A  ->  ( A. x  e.  B  ph 
->  ( x  e.  B  ->  ph ) ) )
1817com23 77 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  -> 
( A. x  e.  B  ph  ->  ph )
) )
199, 15, 18bj-vtoclgft 10736 . . . 4  |-  ( A. x ( x  =  A  ->  ( (
x  e.  B  -> 
( A. x  e.  B  ph  ->  ph )
)  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) ) )  ->  ( A  e.  B  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) ) )
208, 19syl 14 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) ) )
2120pm2.43d 49 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  B  ->  ( A. x  e.  B  ph  ->  ps ) ) )
2221com23 77 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A. x  e.  B  ph  ->  ( A  e.  B  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283    = 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-rspg  10748
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