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Mirrors > Home > ILE Home > Th. List > nfwe | GIF version |
Description: Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfwe.r | ⊢ Ⅎ𝑥𝑅 |
nfwe.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfwe | ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wetr 4264 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfwe.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nfwe.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffr 4279 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
5 | nfcv 2282 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑎 | |
6 | nfcv 2282 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
7 | 5, 2, 6 | nfbr 3982 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
8 | nfcv 2282 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
9 | 6, 2, 8 | nfbr 3982 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
10 | 7, 9 | nfan 1545 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
11 | 5, 2, 8 | nfbr 3982 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
12 | 10, 11 | nfim 1552 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
13 | 3, 12 | nfralxy 2474 | . . . . 5 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 3, 13 | nfralxy 2474 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
15 | 3, 14 | nfralxy 2474 | . . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
16 | 4, 15 | nfan 1545 | . 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 1, 16 | nfxfr 1451 | 1 ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1437 Ⅎwnfc 2269 ∀wral 2417 class class class wbr 3937 Fr wfr 4258 We wwe 4260 |
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-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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 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-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-frfor 4261 df-frind 4262 df-wetr 4264 |
This theorem is referenced by: (None) |
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