Theorem bj-bdfindisg 13146

Description: Version of bj-bdfindis 13145 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13145 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 13145	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
9		bj-bdfindisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
10		nfcv 2281	. . 3 ⊢ Ⅎ𝑥ω
11		bj-bdfindisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
12		bj-bdfindisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
13	9, 10, 11, 12	bj-rspg 12994	. 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 103   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  Ⅎwnfc 2268  ∀wral 2416  ∅c0 3363  suc csuc 4287  ωcom 4504  BOUNDED wbd 13010

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdor 13014  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-infvn 13139

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by:  bj-nntrans  13149  bj-nnelirr  13151  bj-omtrans  13154