Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5515 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
2 | nffr.r | . . . 4 ⊢ Ⅎ𝑥𝑅 | |
3 | nffr.a | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffr 5528 | . . 3 ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 |
5 | 2, 3 | nfso 5479 | . . 3 ⊢ Ⅎ𝑥 𝑅 Or 𝐴 |
6 | 4, 5 | nfan 1896 | . 2 ⊢ Ⅎ𝑥(𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) |
7 | 1, 6 | nfxfr 1849 | 1 ⊢ Ⅎ𝑥 𝑅 We 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1780 Ⅎwnfc 2961 Or wor 5472 Fr wfr 5510 We wwe 5512 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 |
This theorem is referenced by: nfoi 8977 aomclem6 39657 |
Copyright terms: Public domain | W3C validator |