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| Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| nfpo.r | ⊢ Ⅎ𝑥𝑅 | 
| nfpo.a | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| nfso | ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-so 5592 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))) | |
| 2 | nfpo.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
| 3 | nfpo.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nfpo 5597 | . . 3 ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | 
| 5 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
| 6 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
| 7 | 5, 2, 6 | nfbr 5189 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎𝑅𝑏 | 
| 8 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑥 𝑎 = 𝑏 | |
| 9 | 6, 2, 5 | nfbr 5189 | . . . . . 6 ⊢ Ⅎ𝑥 𝑏𝑅𝑎 | 
| 10 | 7, 8, 9 | nf3or 1904 | . . . . 5 ⊢ Ⅎ𝑥(𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) | 
| 11 | 3, 10 | nfralw 3310 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) | 
| 12 | 3, 11 | nfralw 3310 | . . 3 ⊢ Ⅎ𝑥∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) | 
| 13 | 4, 12 | nfan 1898 | . 2 ⊢ Ⅎ𝑥(𝑅 Po 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) | 
| 14 | 1, 13 | nfxfr 1852 | 1 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∨ w3o 1085 Ⅎwnf 1782 Ⅎwnfc 2889 ∀wral 3060 class class class wbr 5142 Po wpo 5589 Or wor 5590 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-po 5591 df-so 5592 | 
| This theorem is referenced by: nfwe 5659 weiunso 36468 | 
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