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Mirrors > Home > ILE Home > Th. List > sbceqg | Unicode version |
Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbceqg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2916 |
. . 3
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2 | dfsbcq2 2916 |
. . . . 5
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3 | 2 | abbidv 2258 |
. . . 4
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4 | dfsbcq2 2916 |
. . . . 5
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5 | 4 | abbidv 2258 |
. . . 4
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6 | 3, 5 | eqeq12d 2155 |
. . 3
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7 | nfs1v 1913 |
. . . . . 6
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8 | 7 | nfab 2287 |
. . . . 5
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9 | nfs1v 1913 |
. . . . . 6
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10 | 9 | nfab 2287 |
. . . . 5
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11 | 8, 10 | nfeq 2290 |
. . . 4
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12 | sbab 2268 |
. . . . 5
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13 | sbab 2268 |
. . . . 5
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14 | 12, 13 | eqeq12d 2155 |
. . . 4
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15 | 11, 14 | sbie 1765 |
. . 3
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16 | 1, 6, 15 | vtoclbg 2750 |
. 2
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17 | df-csb 3008 |
. . 3
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18 | df-csb 3008 |
. . 3
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19 | 17, 18 | eqeq12i 2154 |
. 2
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20 | 16, 19 | syl6bbr 197 |
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-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-csb 3008 |
This theorem is referenced by: sbcne12g 3025 sbceq1g 3027 sbceq2g 3029 sbcfng 5278 |
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