Theorem bj-bdfindisg 16718

Description: Version of bj-bdfindis 16717 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16717 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 16717	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
9		bj-bdfindisg.nfa	. . 3 |- F/_ x A
10		nfcv 2384	. . 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 16559	. 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 104   = wceq 1398   F/wnf 1509   e. wcel 2203   F/_wnfc 2371   A.wral 2520   (/)c0 3508   suc csuc 4486   omcom 4712  BOUNDED wbd 16582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-bd0 16583  ax-bdor 16586  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654  ax-infvn 16711

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by:  bj-nntrans  16721  bj-nnelirr  16723  bj-omtrans  16726