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Theorem bj-bdfindisg 13534
Description: Version of bj-bdfindis 13533 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13533 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 13533 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
9 bj-bdfindisg.nfa . . 3 𝑥𝐴
10 nfcv 2299 . . 3 𝑥ω
11 bj-bdfindisg.nfterm . . 3 𝑥𝜏
12 bj-bdfindisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
139, 10, 11, 12bj-rspg 13372 . 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 103   = wceq 1335  wnf 1440  wcel 2128  wnfc 2286  wral 2435  c0 3394  suc csuc 4325  ωcom 4549  BOUNDED wbd 13398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4090  ax-pr 4169  ax-un 4393  ax-bd0 13399  ax-bdor 13402  ax-bdex 13405  ax-bdeq 13406  ax-bdel 13407  ax-bdsb 13408  ax-bdsep 13470  ax-infvn 13527
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4550  df-bdc 13427  df-bj-ind 13513
This theorem is referenced by:  bj-nntrans  13537  bj-nnelirr  13539  bj-omtrans  13542
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