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Mirrors > Home > ILE Home > Th. List > ralprg | Unicode version |
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralprg.1 | |
ralprg.2 |
Ref | Expression |
---|---|
ralprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3405 | . . . 4 | |
2 | 1 | raleqi 2553 | . . 3 |
3 | ralunb 3153 | . . 3 | |
4 | 2, 3 | bitri 182 | . 2 |
5 | ralprg.1 | . . . 4 | |
6 | 5 | ralsng 3433 | . . 3 |
7 | ralprg.2 | . . . 4 | |
8 | 7 | ralsng 3433 | . . 3 |
9 | 6, 8 | bi2anan9 570 | . 2 |
10 | 4, 9 | syl5bb 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cun 2971 csn 3398 cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-sbc 2816 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: raltpg 3445 ralpr 3447 iinxprg 3752 fvinim0ffz 9250 |
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