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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 2843 |
. . 3
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2 | dfsbcq2 2843 |
. . . . 5
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3 | 2 | abbidv 2205 |
. . . 4
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4 | dfsbcq2 2843 |
. . . . 5
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5 | 4 | abbidv 2205 |
. . . 4
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6 | 3, 5 | eqeq12d 2102 |
. . 3
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7 | nfs1v 1863 |
. . . . . 6
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8 | 7 | nfab 2233 |
. . . . 5
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9 | nfs1v 1863 |
. . . . . 6
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10 | 9 | nfab 2233 |
. . . . 5
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11 | 8, 10 | nfeq 2236 |
. . . 4
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12 | sbab 2214 |
. . . . 5
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13 | sbab 2214 |
. . . . 5
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14 | 12, 13 | eqeq12d 2102 |
. . . 4
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15 | 11, 14 | sbie 1721 |
. . 3
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16 | 1, 6, 15 | vtoclbg 2680 |
. 2
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17 | df-csb 2934 |
. . 3
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18 | df-csb 2934 |
. . 3
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19 | 17, 18 | eqeq12i 2101 |
. 2
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20 | 16, 19 | syl6bbr 196 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-sbc 2841 df-csb 2934 |
This theorem is referenced by: sbcne12g 2949 sbceq1g 2951 sbceq2g 2953 sbcfng 5159 |
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