Theorem bj-findisg 16750

Description: Version of bj-findis 16749 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 16749 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 16749	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
8		bj-findisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
9		nfcv 2384	. . 3 ⊢ Ⅎ𝑥ω
10		bj-findisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
11		bj-findisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
12	8, 9, 10, 11	bj-rspg 16559	. 2 ⊢ (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏))
13	7, 12	syl 14	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  Ⅎwnf 1509   ∈ wcel 2203  Ⅎwnfc 2371  ∀wral 2520  ∅c0 3508  suc csuc 4486  ωcom 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-bd0 16583  ax-bdim 16584  ax-bdan 16585  ax-bdor 16586  ax-bdn 16587  ax-bdal 16588  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654  ax-infvn 16711

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by: (None)