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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 5629 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
2 | nffr.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nffr.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffr 5646 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
5 | 2, 3 | nfso 5590 | . . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
6 | 4, 5 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) |
7 | 1, 6 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 Ⅎwnf 1786 Ⅎwnfc 2884 Or wor 5583 Fr wfr 5624 We wwe 5626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 |
This theorem is referenced by: nfoi 9496 aomclem6 41672 |
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