Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindisg | GIF version |
Description: Version of bj-bdfindis 13533 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13533 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 13533 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
9 | bj-bdfindisg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2299 | . . 3 ⊢ Ⅎ𝑥ω | |
11 | bj-bdfindisg.nfterm | . . 3 ⊢ Ⅎ𝑥𝜏 | |
12 | bj-bdfindisg.term | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) | |
13 | 9, 10, 11, 12 | bj-rspg 13372 | . 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 1335 Ⅎwnf 1440 ∈ wcel 2128 Ⅎwnfc 2286 ∀wral 2435 ∅c0 3394 suc csuc 4325 ωcom 4549 BOUNDED wbd 13398 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-nul 4090 ax-pr 4169 ax-un 4393 ax-bd0 13399 ax-bdor 13402 ax-bdex 13405 ax-bdeq 13406 ax-bdel 13407 ax-bdsb 13408 ax-bdsep 13470 ax-infvn 13527 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-pr 3567 df-uni 3773 df-int 3808 df-suc 4331 df-iom 4550 df-bdc 13427 df-bj-ind 13513 |
This theorem is referenced by: bj-nntrans 13537 bj-nnelirr 13539 bj-omtrans 13542 |
Copyright terms: Public domain | W3C validator |