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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 5643 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
2 | nffr.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nffr.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffr 5662 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
5 | 2, 3 | nfso 5604 | . . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
6 | 4, 5 | nfan 1897 | . 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) |
7 | 1, 6 | nfxfr 1850 | 1 ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 Ⅎwnf 1780 Ⅎwnfc 2888 Or wor 5596 Fr wfr 5638 We wwe 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 |
This theorem is referenced by: nfoi 9552 aomclem6 43048 |
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