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Theorem bj-bdfindisg 13073
Description: Version of bj-bdfindis 13072 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13072 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 13072 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
9 bj-bdfindisg.nfa . . 3  |-  F/_ x A
10 nfcv 2258 . . 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 12921 . 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 1316   F/wnf 1421    e. wcel 1465   F/_wnfc 2245   A.wral 2393   (/)c0 3333   suc csuc 4257   omcom 4474  BOUNDED wbd 12937
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdor 12941  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bj-nntrans  13076  bj-nnelirr  13078  bj-omtrans  13081
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