Theorem bj-bdfindisg 15678

Description: Version of bj-bdfindis 15677 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15677 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 15677	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
9		bj-bdfindisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
10		nfcv 2339	. . 3 ⊢ Ⅎ𝑥ω
11		bj-bdfindisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
12		bj-bdfindisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
13	9, 10, 11, 12	bj-rspg 15517	. 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 1364  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326  ∀wral 2475  ∅c0 3451  suc csuc 4401  ωcom 4627  BOUNDED wbd 15542

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdor 15546  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614  ax-infvn 15671

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by:  bj-nntrans  15681  bj-nnelirr  15683  bj-omtrans  15686