Theorem bj-findisg 15003

Description: Version of bj-findis 15002 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15002 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 15002	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
8		bj-findisg.nfa	. . 3 |- F/_ x A
9		nfcv 2329	. . 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 14810	. 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 1363   F/wnf 1470   e. wcel 2158   F/_wnfc 2316   A.wral 2465   (/)c0 3434   suc csuc 4377   omcom 4601

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-bd0 14836  ax-bdim 14837  ax-bdan 14838  ax-bdor 14839  ax-bdn 14840  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845  ax-bdsep 14907  ax-infvn 14964

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14864  df-bj-ind 14950

This theorem is referenced by: (None)