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Theorem bj-bdfindisg 12957
Description: Version of bj-bdfindis 12956 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 12956 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 12956 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
9 bj-bdfindisg.nfa . . 3 𝑥𝐴
10 nfcv 2256 . . 3 𝑥ω
11 bj-bdfindisg.nfterm . . 3 𝑥𝜏
12 bj-bdfindisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
139, 10, 11, 12bj-rspg 12805 . 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 1314  wnf 1419  wcel 1463  wnfc 2243  wral 2391  c0 3331  suc csuc 4255  ωcom 4472  BOUNDED wbd 12821
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12822  ax-bdor 12825  ax-bdex 12828  ax-bdeq 12829  ax-bdel 12830  ax-bdsb 12831  ax-bdsep 12893  ax-infvn 12950
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12850  df-bj-ind 12936
This theorem is referenced by:  bj-nntrans  12960  bj-nnelirr  12962  bj-omtrans  12965
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