Theorem bj-findisg 17172

Description: Version of bj-findis 17171 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 17171 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-findis.nf0	⊢ Ⅎ𝑥𝜓
bj-findis.nf1	⊢ Ⅎ𝑥𝜒
bj-findis.nfsuc	⊢ Ⅎ𝑥𝜃
bj-findis.0	⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
bj-findis.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
bj-findis.suc	⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
bj-findisg.nfa	⊢ Ⅎ𝑥𝐴
bj-findisg.nfterm	⊢ Ⅎ𝑥𝜏
bj-findisg.term	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))

Assertion

Ref	Expression
bj-findisg	⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (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

Step	Hyp	Ref	Expression
1		bj-findis.nf0	. . 3 ⊢ Ⅎ𝑥𝜓
2		bj-findis.nf1	. . 3 ⊢ Ⅎ𝑥𝜒
3		bj-findis.nfsuc	. . 3 ⊢ Ⅎ𝑥𝜃
4		bj-findis.0	. . 3 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
5		bj-findis.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
6		bj-findis.suc	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
7	1, 2, 3, 4, 5, 6	bj-findis 17171	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
8		bj-findisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
9		nfcv 2392	. . 3 ⊢ Ⅎ𝑥ω
10		bj-findisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
11		bj-findisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
12	8, 9, 10, 11	bj-rspg 16981	. 2 ⊢ (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏))
13	7, 12	syl 14	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  Ⅎwnf 1513   ∈ wcel 2209  Ⅎwnfc 2379  ∀wral 2528  ∅c0 3520  suc csuc 4510  ωcom 4737

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4259  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-bd0 17005  ax-bdim 17006  ax-bdan 17007  ax-bdor 17008  ax-bdn 17009  ax-bdal 17010  ax-bdex 17011  ax-bdeq 17012  ax-bdel 17013  ax-bdsb 17014  ax-bdsep 17076  ax-infvn 17133

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-suc 4516  df-iom 4738  df-bdc 17033  df-bj-ind 17119

This theorem is used by: (None)