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Mirrors > Home > ILE Home > Th. List > raltpg | Unicode version |
Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralprg.1 | |
ralprg.2 | |
raltpg.3 |
Ref | Expression |
---|---|
raltpg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralprg.1 | . . . . 5 | |
2 | ralprg.2 | . . . . 5 | |
3 | 1, 2 | ralprg 3621 | . . . 4 |
4 | raltpg.3 | . . . . 5 | |
5 | 4 | ralsng 3610 | . . . 4 |
6 | 3, 5 | bi2anan9 596 | . . 3 |
7 | 6 | 3impa 1183 | . 2 |
8 | df-tp 3578 | . . . 4 | |
9 | 8 | raleqi 2663 | . . 3 |
10 | ralunb 3298 | . . 3 | |
11 | 9, 10 | bitri 183 | . 2 |
12 | df-3an 969 | . 2 | |
13 | 7, 11, 12 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 cun 3109 csn 3570 cpr 3571 ctp 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-sbc 2947 df-un 3115 df-sn 3576 df-pr 3577 df-tp 3578 |
This theorem is referenced by: raltp 3627 sumtp 11341 |
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