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Theorem bj-findisg 10105
Description: Version of bj-findis 10104 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10104 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 10104 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
8 bj-findisg.nfa . . 3  |-  F/_ x A
9 nfcv 2178 . . 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 9926 . 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 97    = wceq 1243   F/wnf 1349    e. wcel 1393   F/_wnfc 2165   A.wral 2306   (/)c0 3224   suc csuc 4102   omcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-bd0 9933  ax-bdim 9934  ax-bdan 9935  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051
This theorem is referenced by: (None)
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