Theorem bj-bdfindisg 16022

Description: Version of bj-bdfindis 16021 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16021 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 16021	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
9		bj-bdfindisg.nfa	. . 3 |- F/_ x A
10		nfcv 2349	. . 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 15862	. 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 104   = wceq 1373   F/wnf 1484   e. wcel 2177   F/_wnfc 2336   A.wral 2485   (/)c0 3464   suc csuc 4420   omcom 4646  BOUNDED wbd 15886

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 2179  ax-14 2180  ax-ext 2188  ax-nul 4178  ax-pr 4261  ax-un 4488  ax-bd0 15887  ax-bdor 15890  ax-bdex 15893  ax-bdeq 15894  ax-bdel 15895  ax-bdsb 15896  ax-bdsep 15958  ax-infvn 16015

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-suc 4426  df-iom 4647  df-bdc 15915  df-bj-ind 16001

This theorem is referenced by:  bj-nntrans  16025  bj-nnelirr  16027  bj-omtrans  16030