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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindisg | GIF version |
Description: Version of bj-bdfindis 13134 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13134 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 13134 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
9 | bj-bdfindisg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2279 | . . 3 ⊢ Ⅎ𝑥ω | |
11 | bj-bdfindisg.nfterm | . . 3 ⊢ Ⅎ𝑥𝜏 | |
12 | bj-bdfindisg.term | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) | |
13 | 9, 10, 11, 12 | bj-rspg 12983 | . 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 1331 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2266 ∀wral 2414 ∅c0 3358 suc csuc 4282 ωcom 4499 BOUNDED wbd 12999 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-bd0 13000 ax-bdor 13003 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 ax-infvn 13128 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: bj-nntrans 13138 bj-nnelirr 13140 bj-omtrans 13143 |
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