Theorem bj-bdfindisg 15510

Description: Version of bj-bdfindis 15509 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15509 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 15509	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
9		bj-bdfindisg.nfa	. . 3 |- F/_ x A
10		nfcv 2336	. . 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 15349	. 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 1364   F/wnf 1471   e. wcel 2164   F/_wnfc 2323   A.wral 2472   (/)c0 3447   suc csuc 4397   omcom 4623  BOUNDED wbd 15374

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4156  ax-pr 4239  ax-un 4465  ax-bd0 15375  ax-bdor 15378  ax-bdex 15381  ax-bdeq 15382  ax-bdel 15383  ax-bdsb 15384  ax-bdsep 15446  ax-infvn 15503

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-suc 4403  df-iom 4624  df-bdc 15403  df-bj-ind 15489

This theorem is referenced by:  bj-nntrans  15513  bj-nnelirr  15515  bj-omtrans  15518