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Mirrors > Home > MPE Home > Th. List > nfwe | Structured version Visualization version 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 |
---|---|
nffr.r | ⊢ Ⅎ𝑥𝑅 |
nffr.a | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfwe | ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-we 5518 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
2 | nffr.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nffr.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffr 5531 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
5 | 2, 3 | nfso 5482 | . . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
6 | 4, 5 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) |
7 | 1, 6 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1784 Ⅎwnfc 2963 Or wor 5475 Fr wfr 5513 We wwe 5515 |
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-3or 1084 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-rex 3146 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 df-so 5477 df-fr 5516 df-we 5518 |
This theorem is referenced by: nfoi 8980 aomclem6 39666 |
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