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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 5639 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
| 2 | nffr.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
| 3 | nffr.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nffr 5658 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
| 5 | 2, 3 | nfso 5599 | . . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
| 6 | 4, 5 | nfan 1899 | . 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 395 Ⅎwnf 1783 Ⅎwnfc 2890 Or wor 5591 Fr wfr 5634 We wwe 5636 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 |
| This theorem is referenced by: nfoi 9554 aomclem6 43071 |
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