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Mirrors > Home > MPE Home > Th. List > csbif | Structured version Visualization version Unicode version |
Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.) |
Ref | Expression |
---|---|
csbif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3377 |
. . . 4
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2 | dfsbcq2 3281 |
. . . . 5
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3 | csbeq1 3377 |
. . . . 5
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4 | csbeq1 3377 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 2, 3, 4 | ifbieq12d 3919 |
. . . 4
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6 | 1, 5 | eqeq12d 2476 |
. . 3
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7 | vex 3059 |
. . . 4
![]() ![]() ![]() ![]() | |
8 | nfs1v 2276 |
. . . . 5
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9 | nfcsb1v 3390 |
. . . . 5
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10 | nfcsb1v 3390 |
. . . . 5
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11 | 8, 9, 10 | nfif 3921 |
. . . 4
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12 | sbequ12 2093 |
. . . . 5
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13 | csbeq1a 3383 |
. . . . 5
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14 | csbeq1a 3383 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 12, 13, 14 | ifbieq12d 3919 |
. . . 4
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16 | 7, 11, 15 | csbief 3399 |
. . 3
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17 | 6, 16 | vtoclg 3118 |
. 2
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18 | csbprc 3781 |
. . 3
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19 | csbprc 3781 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | csbprc 3781 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | ifeq12d 3912 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | ifid 3929 |
. . . 4
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23 | 21, 22 | syl6req 2512 |
. . 3
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24 | 18, 23 | eqtrd 2495 |
. 2
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25 | 17, 24 | pm2.61i 169 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-fal 1460 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 |
This theorem is referenced by: fvmptnn04if 19921 csbopg2 31769 csbrdgg 31774 csbfinxpg 31824 cdlemk40 34528 |
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