Theorem bj-bdfindisg 16311

Description: Version of bj-bdfindis 16310 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16310 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 16310	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
9		bj-bdfindisg.nfa	. . 3 ⊢ Ⅎ𝑥𝐴
10		nfcv 2372	. . 3 ⊢ Ⅎ𝑥ω
11		bj-bdfindisg.nfterm	. . 3 ⊢ Ⅎ𝑥𝜏
12		bj-bdfindisg.term	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))
13	9, 10, 11, 12	bj-rspg 16151	. 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 1395  Ⅎwnf 1506   ∈ wcel 2200  Ⅎwnfc 2359  ∀wral 2508  ∅c0 3491  suc csuc 4456  ωcom 4682  BOUNDED wbd 16175

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4210  ax-pr 4293  ax-un 4524  ax-bd0 16176  ax-bdor 16179  ax-bdex 16182  ax-bdeq 16183  ax-bdel 16184  ax-bdsb 16185  ax-bdsep 16247  ax-infvn 16304

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-suc 4462  df-iom 4683  df-bdc 16204  df-bj-ind 16290

This theorem is referenced by:  bj-nntrans  16314  bj-nnelirr  16316  bj-omtrans  16319