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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 2921 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | nfnfc1 2285 |
. . 3
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4 | id 19 |
. . . 4
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5 | nfcvd 2283 |
. . . 4
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6 | 4, 5 | nfeld 2298 |
. . 3
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7 | sbcex 2921 |
. . . 4
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8 | 7 | 2a1i 27 |
. . 3
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9 | 3, 6, 8 | rexlimd2 2550 |
. 2
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10 | sbcco 2934 |
. . . 4
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11 | simpl 108 |
. . . . 5
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12 | sbsbc 2917 |
. . . . . . 7
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13 | nfcv 2282 |
. . . . . . . . 9
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14 | nfs1v 1913 |
. . . . . . . . 9
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15 | 13, 14 | nfrexxy 2475 |
. . . . . . . 8
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16 | sbequ12 1745 |
. . . . . . . . 9
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17 | 16 | rexbidv 2439 |
. . . . . . . 8
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18 | 15, 17 | sbie 1765 |
. . . . . . 7
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19 | 12, 18 | bitr3i 185 |
. . . . . 6
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20 | nfcvd 2283 |
. . . . . . . . . 10
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21 | 20, 4 | nfeqd 2297 |
. . . . . . . . 9
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22 | 3, 21 | nfan1 1544 |
. . . . . . . 8
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23 | dfsbcq2 2916 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | adantl 275 |
. . . . . . . 8
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25 | 22, 24 | rexbid 2437 |
. . . . . . 7
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26 | 25 | adantll 468 |
. . . . . 6
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27 | 19, 26 | syl5bb 191 |
. . . . 5
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28 | 11, 27 | sbcied 2949 |
. . . 4
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29 | 10, 28 | bitr3id 193 |
. . 3
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30 | 29 | expcom 115 |
. 2
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31 | 2, 9, 30 | pm5.21ndd 695 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 |
This theorem is referenced by: sbcrex 2992 |
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