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Mirrors > Home > MPE Home > Th. List > nfpo | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
nfpo.r | ⊢ Ⅎ𝑥𝑅 |
nfpo.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfpo | ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-po 5476 | . 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐))) | |
2 | nfpo.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2979 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑎 | |
4 | nfpo.r | . . . . . . . 8 ⊢ Ⅎ𝑥𝑅 | |
5 | 3, 4, 3 | nfbr 5115 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑎 |
6 | 5 | nfn 1857 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝑎𝑅𝑎 |
7 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑏 | |
8 | 3, 4, 7 | nfbr 5115 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 |
9 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝑐 | |
10 | 7, 4, 9 | nfbr 5115 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑏𝑅𝑐 |
11 | 8, 10 | nfan 1900 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) |
12 | 3, 4, 9 | nfbr 5115 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑎𝑅𝑐 |
13 | 11, 12 | nfim 1897 | . . . . . 6 ⊢ Ⅎ𝑥((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐) |
14 | 6, 13 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑥(¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
15 | 2, 14 | nfralw 3227 | . . . 4 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
16 | 2, 15 | nfralw 3227 | . . 3 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
17 | 2, 16 | nfralw 3227 | . 2 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 (¬ 𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏 ∧ 𝑏𝑅𝑐) → 𝑎𝑅𝑐)) |
18 | 1, 17 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 Ⅎwnf 1784 Ⅎwnfc 2963 ∀wral 3140 class class class wbr 5068 Po wpo 5474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-po 5476 |
This theorem is referenced by: nfso 5482 |
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