Theorem bj-findisg 15916

Description: Version of bj-findis 15915 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15915 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 15915	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
8		bj-findisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
9		nfcv 2348	. . 3 ⊢ Ⅎ𝑥ω
10		bj-findisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
11		bj-findisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
12	8, 9, 10, 11	bj-rspg 15723	. 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 1373  Ⅎwnf 1483   ∈ wcel 2176  Ⅎwnfc 2335  ∀wral 2484  ∅c0 3460  suc csuc 4412  ωcom 4638

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-bd0 15749  ax-bdim 15750  ax-bdan 15751  ax-bdor 15752  ax-bdn 15753  ax-bdal 15754  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by: (None)