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Theorem bj-bdfindisg 13983
Description: Version of bj-bdfindis 13982 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13982 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 13982 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
9 bj-bdfindisg.nfa . . 3 𝑥𝐴
10 nfcv 2312 . . 3 𝑥ω
11 bj-bdfindisg.nfterm . . 3 𝑥𝜏
12 bj-bdfindisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
139, 10, 11, 12bj-rspg 13822 . 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 1348  wnf 1453  wcel 2141  wnfc 2299  wral 2448  c0 3414  suc csuc 4350  ωcom 4574  BOUNDED wbd 13847
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-infvn 13976
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962
This theorem is referenced by:  bj-nntrans  13986  bj-nnelirr  13988  bj-omtrans  13991
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