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Mirrors > Home > ILE Home > Th. List > ralxfrd | Unicode version |
Description: Transfer universal quantification from a variable to another variable contained in expression . (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
ralxfrd.1 | |
ralxfrd.2 | |
ralxfrd.3 |
Ref | Expression |
---|---|
ralxfrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxfrd.1 | . . . 4 | |
2 | ralxfrd.3 | . . . . 5 | |
3 | 2 | adantlr 446 | . . . 4 |
4 | 1, 3 | rspcdv 2659 | . . 3 |
5 | 4 | ralrimdva 2399 | . 2 |
6 | ralxfrd.2 | . . . 4 | |
7 | r19.29 2450 | . . . . 5 | |
8 | 2 | biimprd 147 | . . . . . . . . 9 |
9 | 8 | expimpd 345 | . . . . . . . 8 |
10 | 9 | ancomsd 256 | . . . . . . 7 |
11 | 10 | ad2antrr 457 | . . . . . 6 |
12 | 11 | rexlimdva 2433 | . . . . 5 |
13 | 7, 12 | syl5 28 | . . . 4 |
14 | 6, 13 | mpan2d 404 | . . 3 |
15 | 14 | ralrimdva 2399 | . 2 |
16 | 5, 15 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 |
This theorem is referenced by: ralxfr2d 4196 ralxfr 4198 |
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