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Theorem bj-findisg 15135
Description: Version of bj-findis 15134 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15134 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-findis.nf0  |-  F/ x ps
bj-findis.nf1  |-  F/ x ch
bj-findis.nfsuc  |-  F/ x th
bj-findis.0  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
bj-findis.1  |-  ( x  =  y  ->  ( ph  ->  ch ) )
bj-findis.suc  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
bj-findisg.nfa  |-  F/_ x A
bj-findisg.nfterm  |-  F/ x ta
bj-findisg.term  |-  ( x  =  A  ->  ( ph  ->  ta ) )
Assertion
Ref Expression
bj-findisg  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  ( A  e.  om  ->  ta ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    ch( x, y)    th( x, y)    ta( x, y)    A( x, y)

Proof of Theorem bj-findisg
StepHypRef Expression
1 bj-findis.nf0 . . 3  |-  F/ x ps
2 bj-findis.nf1 . . 3  |-  F/ x ch
3 bj-findis.nfsuc . . 3  |-  F/ x th
4 bj-findis.0 . . 3  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
5 bj-findis.1 . . 3  |-  ( x  =  y  ->  ( ph  ->  ch ) )
6 bj-findis.suc . . 3  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
71, 2, 3, 4, 5, 6bj-findis 15134 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
8 bj-findisg.nfa . . 3  |-  F/_ x A
9 nfcv 2332 . . 3  |-  F/_ x om
10 bj-findisg.nfterm . . 3  |-  F/ x ta
11 bj-findisg.term . . 3  |-  ( x  =  A  ->  ( ph  ->  ta ) )
128, 9, 10, 11bj-rspg 14942 . 2  |-  ( A. x  e.  om  ph  ->  ( A  e.  om  ->  ta ) )
137, 12syl 14 1  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  ( A  e.  om  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364   F/wnf 1471    e. wcel 2160   F/_wnfc 2319   A.wral 2468   (/)c0 3437   suc csuc 4380   omcom 4604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-bd0 14968  ax-bdim 14969  ax-bdan 14970  ax-bdor 14971  ax-bdn 14972  ax-bdal 14973  ax-bdex 14974  ax-bdeq 14975  ax-bdel 14976  ax-bdsb 14977  ax-bdsep 15039  ax-infvn 15096
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 14996  df-bj-ind 15082
This theorem is referenced by: (None)
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