Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-findisg Unicode version

Theorem bj-findisg 10492
Description: Version of bj-findis 10491 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10491 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 10491 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
8 bj-findisg.nfa . . 3  |-  F/_ x A
9 nfcv 2194 . . 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 10313 . 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 101    = wceq 1259   F/wnf 1365    e. wcel 1409   F/_wnfc 2181   A.wral 2323   (/)c0 3252   suc csuc 4130   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-bd0 10320  ax-bdim 10321  ax-bdan 10322  ax-bdor 10323  ax-bdn 10324  ax-bdal 10325  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-infvn 10453
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator