Theorem bj-findisg 14592

Description: Version of bj-findis 14591 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 14591 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 14591	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
8		bj-findisg.nfa	. . 3 |- F/_ x A
9		nfcv 2319	. . 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 14399	. 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 1353   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   A.wral 2455   (/)c0 3422   suc csuc 4364   omcom 4588

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4128  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-bd0 14425  ax-bdim 14426  ax-bdan 14427  ax-bdor 14428  ax-bdn 14429  ax-bdal 14430  ax-bdex 14431  ax-bdeq 14432  ax-bdel 14433  ax-bdsb 14434  ax-bdsep 14496  ax-infvn 14553

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4370  df-iom 4589  df-bdc 14453  df-bj-ind 14539

This theorem is referenced by: (None)