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Theorem bj-findisg 10918
Description: Version of bj-findis 10917 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10917 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-findis.nf0 𝑥𝜓
bj-findis.nf1 𝑥𝜒
bj-findis.nfsuc 𝑥𝜃
bj-findis.0 (𝑥 = ∅ → (𝜓𝜑))
bj-findis.1 (𝑥 = 𝑦 → (𝜑𝜒))
bj-findis.suc (𝑥 = suc 𝑦 → (𝜃𝜑))
bj-findisg.nfa 𝑥𝐴
bj-findisg.nfterm 𝑥𝜏
bj-findisg.term (𝑥 = 𝐴 → (𝜑𝜏))
Assertion
Ref Expression
bj-findisg ((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-findisg
StepHypRef Expression
1 bj-findis.nf0 . . 3 𝑥𝜓
2 bj-findis.nf1 . . 3 𝑥𝜒
3 bj-findis.nfsuc . . 3 𝑥𝜃
4 bj-findis.0 . . 3 (𝑥 = ∅ → (𝜓𝜑))
5 bj-findis.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
6 bj-findis.suc . . 3 (𝑥 = suc 𝑦 → (𝜃𝜑))
71, 2, 3, 4, 5, 6bj-findis 10917 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
8 bj-findisg.nfa . . 3 𝑥𝐴
9 nfcv 2220 . . 3 𝑥ω
10 bj-findisg.nfterm . . 3 𝑥𝜏
11 bj-findisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
128, 9, 10, 11bj-rspg 10733 . 2 (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏))
137, 12syl 14 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wnf 1390  wcel 1434  wnfc 2207  wral 2349  c0 3252  suc csuc 4122  ωcom 4333
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3906  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-bd0 10747  ax-bdim 10748  ax-bdan 10749  ax-bdor 10750  ax-bdn 10751  ax-bdal 10752  ax-bdex 10753  ax-bdeq 10754  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818  ax-infvn 10879
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-pr 3407  df-uni 3604  df-int 3639  df-suc 4128  df-iom 4334  df-bdc 10775  df-bj-ind 10865
This theorem is referenced by: (None)
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