Theorem bj-bdfindisg 16543

Description: Version of bj-bdfindis 16542 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16542 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-bdfindisg	⊢ ((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-bdfindisg

Step	Hyp	Ref	Expression
1		bj-bdfindis.bd	. . 3 ⊢ BOUNDED 𝜑
2		bj-bdfindis.nf0	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-bdfindis.nf1	. . 3 ⊢ Ⅎ𝑥𝜒
4		bj-bdfindis.nfsuc	. . 3 ⊢ Ⅎ𝑥𝜃
5		bj-bdfindis.0	. . 3 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
6		bj-bdfindis.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7		bj-bdfindis.suc	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
8	1, 2, 3, 4, 5, 6, 7	bj-bdfindis 16542	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
9		bj-bdfindisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
10		nfcv 2374	. . 3 ⊢ Ⅎ𝑥ω
11		bj-bdfindisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
12		bj-bdfindisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
13	9, 10, 11, 12	bj-rspg 16383	. 2 ⊢ (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏))
14	8, 13	syl 14	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  Ⅎwnf 1508   ∈ wcel 2202  Ⅎwnfc 2361  ∀wral 2510  ∅c0 3494  suc csuc 4462  ωcom 4688  BOUNDED wbd 16407

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4215  ax-pr 4299  ax-un 4530  ax-bd0 16408  ax-bdor 16411  ax-bdex 16414  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417  ax-bdsep 16479  ax-infvn 16536

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689  df-bdc 16436  df-bj-ind 16522

This theorem is referenced by:  bj-nntrans  16546  bj-nnelirr  16548  bj-omtrans  16551