Theorem bj-findisg 16115

Description: Version of bj-findis 16114 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 16114 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 16114	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
8		bj-findisg.nfa	. . 3 |- F/_ x A
9		nfcv 2350	. . 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 15923	. 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 1373   F/wnf 1484   e. wcel 2178   F/_wnfc 2337   A.wral 2486   (/)c0 3468   suc csuc 4430   omcom 4656

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-nul 4186  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-bd0 15948  ax-bdim 15949  ax-bdan 15950  ax-bdor 15951  ax-bdn 15952  ax-bdal 15953  ax-bdex 15954  ax-bdeq 15955  ax-bdel 15956  ax-bdsb 15957  ax-bdsep 16019  ax-infvn 16076

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657  df-bdc 15976  df-bj-ind 16062

This theorem is referenced by: (None)