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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindisg | GIF version |
Description: Version of bj-bdfindis 13316 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13316 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 13316 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
9 | bj-bdfindisg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2282 | . . 3 ⊢ Ⅎ𝑥ω | |
11 | bj-bdfindisg.nfterm | . . 3 ⊢ Ⅎ𝑥𝜏 | |
12 | bj-bdfindisg.term | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) | |
13 | 9, 10, 11, 12 | bj-rspg 13165 | . 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 103 = wceq 1332 Ⅎwnf 1437 ∈ wcel 1481 Ⅎwnfc 2269 ∀wral 2417 ∅c0 3368 suc csuc 4295 ωcom 4512 BOUNDED wbd 13181 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-bd0 13182 ax-bdor 13185 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 ax-infvn 13310 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: bj-nntrans 13320 bj-nnelirr 13322 bj-omtrans 13325 |
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