Theorem bj-bdfindisg 14861

Description: Version of bj-bdfindis 14860 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 14860 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 14860	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
9		bj-bdfindisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
10		nfcv 2319	. . 3 ⊢ Ⅎ𝑥ω
11		bj-bdfindisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
12		bj-bdfindisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
13	9, 10, 11, 12	bj-rspg 14700	. 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 1353  Ⅎwnf 1460   ∈ wcel 2148  Ⅎwnfc 2306  ∀wral 2455  ∅c0 3424  suc csuc 4367  ωcom 4591  BOUNDED wbd 14725

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14726  ax-bdor 14729  ax-bdex 14732  ax-bdeq 14733  ax-bdel 14734  ax-bdsb 14735  ax-bdsep 14797  ax-infvn 14854

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14754  df-bj-ind 14840

This theorem is referenced by:  bj-nntrans  14864  bj-nnelirr  14866  bj-omtrans  14869