Theorem bj-bdfindisg 13317

Description: Version of bj-bdfindis 13316 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13316 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 13316	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
9		bj-bdfindisg.nfa	. . 3 |- F/_ x A
10		nfcv 2282	. . 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 13165	. 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 1332   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   A.wral 2417   (/)c0 3368   suc csuc 4295   omcom 4512  BOUNDED wbd 13181

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062  ax-pr 4139  ax-un 4363  ax-bd0 13182  ax-bdor 13185  ax-bdex 13188  ax-bdeq 13189  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253  ax-infvn 13310

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513  df-bdc 13210  df-bj-ind 13296

This theorem is referenced by:  bj-nntrans  13320  bj-nnelirr  13322  bj-omtrans  13325