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

Step	Hyp	Ref	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 ) )
8	1, 2, 3, 4, 5, 6, 7	bj-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 ) )
13	9, 10, 11, 12	bj-rspg 11953	. 2 |- ( A. x e. om ph -> ( A e. om -> ta ) )
14	8, 13	syl 14	1 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )

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