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Theorem bj-bdfindisg 11726
Description: Version of bj-bdfindis 11725 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 11725 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd BOUNDED 𝜑
bj-bdfindis.nf0 𝑥𝜓
bj-bdfindis.nf1 𝑥𝜒
bj-bdfindis.nfsuc 𝑥𝜃
bj-bdfindis.0 (𝑥 = ∅ → (𝜓𝜑))
bj-bdfindis.1 (𝑥 = 𝑦 → (𝜑𝜒))
bj-bdfindis.suc (𝑥 = suc 𝑦 → (𝜃𝜑))
bj-bdfindisg.nfa 𝑥𝐴
bj-bdfindisg.nfterm 𝑥𝜏
bj-bdfindisg.term (𝑥 = 𝐴 → (𝜑𝜏))
Assertion
Ref Expression
bj-bdfindisg ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem bj-bdfindisg
StepHypRef Expression
1 bj-bdfindis.bd . . 3 BOUNDED 𝜑
2 bj-bdfindis.nf0 . . 3 𝑥𝜓
3 bj-bdfindis.nf1 . . 3 𝑥𝜒
4 bj-bdfindis.nfsuc . . 3 𝑥𝜃
5 bj-bdfindis.0 . . 3 (𝑥 = ∅ → (𝜓𝜑))
6 bj-bdfindis.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
7 bj-bdfindis.suc . . 3 (𝑥 = suc 𝑦 → (𝜃𝜑))
81, 2, 3, 4, 5, 6, 7bj-bdfindis 11725 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
9 bj-bdfindisg.nfa . . 3 𝑥𝐴
10 nfcv 2228 . . 3 𝑥ω
11 bj-bdfindisg.nfterm . . 3 𝑥𝜏
12 bj-bdfindisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
139, 10, 11, 12bj-rspg 11570 . 2 (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏))
148, 13syl 14 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  wral 2359  c0 3286  suc csuc 4190  ωcom 4403  BOUNDED wbd 11586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963  ax-pr 4034  ax-un 4258  ax-bd0 11587  ax-bdor 11590  ax-bdex 11593  ax-bdeq 11594  ax-bdel 11595  ax-bdsb 11596  ax-bdsep 11658  ax-infvn 11719
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3450  df-pr 3451  df-uni 3652  df-int 3687  df-suc 4196  df-iom 4404  df-bdc 11615  df-bj-ind 11705
This theorem is referenced by:  bj-nntrans  11729  bj-nnelirr  11731  bj-omtrans  11734
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