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Mirrors > Home > MPE Home > Th. List > ralprg | Structured version Visualization version 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 4180 | . . . 4 | |
2 | 1 | raleqi 3142 | . . 3 |
3 | ralunb 3794 | . . 3 | |
4 | 2, 3 | bitri 264 | . 2 |
5 | ralprg.1 | . . . 4 | |
6 | 5 | ralsng 4218 | . . 3 |
7 | ralprg.2 | . . . 4 | |
8 | 7 | ralsng 4218 | . . 3 |
9 | 6, 8 | bi2anan9 917 | . 2 |
10 | 4, 9 | syl5bb 272 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cun 3572 csn 4177 cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-sbc 3436 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: raltpg 4236 ralpr 4238 iinxprg 4601 disjprg 4648 fpropnf1 6524 f12dfv 6529 f13dfv 6530 suppr 8377 infpr 8409 sumpr 14477 gcdcllem2 15222 lcmfpr 15340 joinval2lem 17008 meetval2lem 17022 sgrp2rid2 17413 sgrp2nmndlem4 17415 sgrp2nmndlem5 17416 iccntr 22624 limcun 23659 cplgr3v 26331 3wlkdlem4 27022 frgr3v 27139 3vfriswmgr 27142 prsiga 30194 pfx2 41412 |
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