Theorem bj-findisg 15190

Description: Version of bj-findis 15189 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15189 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

Step	Hyp	Ref	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 ) )
7	1, 2, 3, 4, 5, 6	bj-findis 15189	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
8		bj-findisg.nfa	. . 3 |- F/_ x A
9		nfcv 2332	. . 3 |- F/_ x om
10		bj-findisg.nfterm	. . 3 |- F/ x ta
11		bj-findisg.term	. . 3 |- ( x = A -> ( ph -> ta ) )
12	8, 9, 10, 11	bj-rspg 14997	. 2 |- ( A. x e. om ph -> ( A e. om -> ta ) )
13	7, 12	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 2160   F/_wnfc 2319   A.wral 2468   (/)c0 3437   suc csuc 4383   omcom 4607

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 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-bd0 15023  ax-bdim 15024  ax-bdan 15025  ax-bdor 15026  ax-bdn 15027  ax-bdal 15028  ax-bdex 15029  ax-bdeq 15030  ax-bdel 15031  ax-bdsb 15032  ax-bdsep 15094  ax-infvn 15151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608  df-bdc 15051  df-bj-ind 15137

This theorem is referenced by: (None)