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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindisg | GIF version |
Description: Version of bj-bdfindis 10742 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10742 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
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)) |
Ref | Expression |
---|---|
bj-bdfindisg | ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) |
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 10742 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
9 | bj-bdfindisg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2219 | . . 3 ⊢ Ⅎ𝑥ω | |
11 | bj-bdfindisg.nfterm | . . 3 ⊢ Ⅎ𝑥𝜏 | |
12 | bj-bdfindisg.term | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) | |
13 | 9, 10, 11, 12 | bj-rspg 10597 | . 2 ⊢ (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏)) |
14 | 8, 13 | syl 14 | 1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Ⅎwnfc 2206 ∀wral 2348 ∅c0 3251 suc csuc 4120 ωcom 4331 BOUNDED wbd 10603 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 ax-pr 3964 ax-un 4188 ax-bd0 10604 ax-bdor 10607 ax-bdex 10610 ax-bdeq 10611 ax-bdel 10612 ax-bdsb 10613 ax-bdsep 10675 ax-infvn 10736 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 df-bdc 10632 df-bj-ind 10722 |
This theorem is referenced by: bj-nntrans 10746 bj-nnelirr 10748 bj-omtrans 10751 |
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