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Theorem bj-findisg 13755
Description: Version of bj-findis 13754 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13754 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 13754 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
8 bj-findisg.nfa . . 3 𝑥𝐴
9 nfcv 2306 . . 3 𝑥ω
10 bj-findisg.nfterm . . 3 𝑥𝜏
11 bj-findisg.term . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
128, 9, 10, 11bj-rspg 13561 . 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 103   = wceq 1342  wnf 1447  wcel 2135  wnfc 2293  wral 2442  c0 3407  suc csuc 4340  ωcom 4564
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-nul 4105  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-bd0 13588  ax-bdim 13589  ax-bdan 13590  ax-bdor 13591  ax-bdn 13592  ax-bdal 13593  ax-bdex 13594  ax-bdeq 13595  ax-bdel 13596  ax-bdsb 13597  ax-bdsep 13659  ax-infvn 13716
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-sn 3579  df-pr 3580  df-uni 3787  df-int 3822  df-suc 4346  df-iom 4565  df-bdc 13616  df-bj-ind 13702
This theorem is referenced by: (None)
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