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Mirrors > Home > ILE Home > Th. List > sbcrext | Unicode version |
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbcrext |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2870 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | nfnfc1 2243 |
. . 3
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4 | id 19 |
. . . 4
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5 | nfcvd 2241 |
. . . 4
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6 | 4, 5 | nfeld 2256 |
. . 3
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7 | sbcex 2870 |
. . . 4
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8 | 7 | 2a1i 27 |
. . 3
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9 | 3, 6, 8 | rexlimd2 2506 |
. 2
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10 | sbcco 2883 |
. . . 4
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11 | simpl 108 |
. . . . 5
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12 | sbsbc 2866 |
. . . . . . 7
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13 | nfcv 2240 |
. . . . . . . . 9
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14 | nfs1v 1875 |
. . . . . . . . 9
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15 | 13, 14 | nfrexxy 2431 |
. . . . . . . 8
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16 | sbequ12 1712 |
. . . . . . . . 9
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17 | 16 | rexbidv 2397 |
. . . . . . . 8
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18 | 15, 17 | sbie 1732 |
. . . . . . 7
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19 | 12, 18 | bitr3i 185 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | nfcvd 2241 |
. . . . . . . . . 10
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21 | 20, 4 | nfeqd 2255 |
. . . . . . . . 9
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22 | 3, 21 | nfan1 1511 |
. . . . . . . 8
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23 | dfsbcq2 2865 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | adantl 273 |
. . . . . . . 8
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25 | 22, 24 | rexbid 2395 |
. . . . . . 7
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26 | 25 | adantll 463 |
. . . . . 6
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27 | 19, 26 | syl5bb 191 |
. . . . 5
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28 | 11, 27 | sbcied 2897 |
. . . 4
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29 | 10, 28 | syl5bbr 193 |
. . 3
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30 | 29 | expcom 115 |
. 2
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31 | 2, 9, 30 | pm5.21ndd 662 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 |
This theorem is referenced by: sbcrex 2940 |
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