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Theorem bj-bdfindisg 12109
Description: Version of bj-bdfindis 12108 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 12108 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd  |- BOUNDED  ph
bj-bdfindis.nf0  |-  F/ x ps
bj-bdfindis.nf1  |-  F/ x ch
bj-bdfindis.nfsuc  |-  F/ x th
bj-bdfindis.0  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
bj-bdfindis.1  |-  ( x  =  y  ->  ( ph  ->  ch ) )
bj-bdfindis.suc  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
bj-bdfindisg.nfa  |-  F/_ x A
bj-bdfindisg.nfterm  |-  F/ x ta
bj-bdfindisg.term  |-  ( x  =  A  ->  ( ph  ->  ta ) )
Assertion
Ref Expression
bj-bdfindisg  |-  ( ( 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-bdfindisg
StepHypRef Expression
1 bj-bdfindis.bd . . 3  |- BOUNDED  ph
2 bj-bdfindis.nf0 . . 3  |-  F/ x ps
3 bj-bdfindis.nf1 . . 3  |-  F/ x ch
4 bj-bdfindis.nfsuc . . 3  |-  F/ x th
5 bj-bdfindis.0 . . 3  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
6 bj-bdfindis.1 . . 3  |-  ( x  =  y  ->  ( ph  ->  ch ) )
7 bj-bdfindis.suc . . 3  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
81, 2, 3, 4, 5, 6, 7bj-bdfindis 12108 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
9 bj-bdfindisg.nfa . . 3  |-  F/_ x A
10 nfcv 2229 . . 3  |-  F/_ x om
11 bj-bdfindisg.nfterm . . 3  |-  F/ x ta
12 bj-bdfindisg.term . . 3  |-  ( x  =  A  ->  ( ph  ->  ta ) )
139, 10, 11, 12bj-rspg 11953 . 2  |-  ( A. x  e.  om  ph  ->  ( A  e.  om  ->  ta ) )
148, 13syl 14 1  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  ( A  e.  om  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290   F/wnf 1395    e. wcel 1439   F/_wnfc 2216   A.wral 2360   (/)c0 3287   suc csuc 4201   omcom 4418  BOUNDED wbd 11969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-nul 3971  ax-pr 4045  ax-un 4269  ax-bd0 11970  ax-bdor 11973  ax-bdex 11976  ax-bdeq 11977  ax-bdel 11978  ax-bdsb 11979  ax-bdsep 12041  ax-infvn 12102
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-suc 4207  df-iom 4419  df-bdc 11998  df-bj-ind 12088
This theorem is referenced by:  bj-nntrans  12112  bj-nnelirr  12114  bj-omtrans  12117
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