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Theorem bj-bdfindisg 10073
Description: Version of bj-bdfindis 10072 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 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 10072 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
9 bj-bdfindisg.nfa . . 3 𝑥𝐴
10 nfcv 2178 . . 3 𝑥ω
11 bj-bdfindisg.nfterm . . 3 𝑥𝜏
12 bj-bdfindisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
139, 10, 11, 12bj-rspg 9926 . 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 97   = wceq 1243  wnf 1349  wcel 1393  wnfc 2165  wral 2306  c0 3224  suc csuc 4102  ωcom 4313  BOUNDED wbd 9932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051
This theorem is referenced by:  bj-nntrans  10076  bj-nnelirr  10078  bj-omtrans  10081
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