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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 2986 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | nfnfc1 2335 |
. . 3
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4 | id 19 |
. . . 4
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5 | nfcvd 2333 |
. . . 4
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6 | 4, 5 | nfeld 2348 |
. . 3
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7 | sbcex 2986 |
. . . 4
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8 | 7 | 2a1i 27 |
. . 3
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9 | 3, 6, 8 | rexlimd2 2605 |
. 2
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10 | sbcco 2999 |
. . . 4
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11 | simpl 109 |
. . . . 5
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12 | sbsbc 2981 |
. . . . . . 7
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13 | nfcv 2332 |
. . . . . . . . 9
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14 | nfs1v 1951 |
. . . . . . . . 9
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15 | 13, 14 | nfrexxy 2529 |
. . . . . . . 8
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16 | sbequ12 1782 |
. . . . . . . . 9
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17 | 16 | rexbidv 2491 |
. . . . . . . 8
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18 | 15, 17 | sbie 1802 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 12, 18 | bitr3i 186 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | nfcvd 2333 |
. . . . . . . . . 10
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21 | 20, 4 | nfeqd 2347 |
. . . . . . . . 9
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22 | 3, 21 | nfan1 1575 |
. . . . . . . 8
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23 | dfsbcq2 2980 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | adantl 277 |
. . . . . . . 8
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25 | 22, 24 | rexbid 2489 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | adantll 476 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 19, 26 | bitrid 192 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 11, 27 | sbcied 3014 |
. . . 4
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29 | 10, 28 | bitr3id 194 |
. . 3
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30 | 29 | expcom 116 |
. 2
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31 | 2, 9, 30 | pm5.21ndd 706 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 |
This theorem is referenced by: sbcrex 3057 |
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