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Mirrors > Home > ILE Home > Th. List > sbcbrg | Unicode version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
sbcbrg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2879 |
. 2
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2 | csbeq1 2972 |
. . 3
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3 | csbeq1 2972 |
. . 3
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4 | csbeq1 2972 |
. . 3
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5 | 2, 3, 4 | breq123d 3907 |
. 2
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6 | nfcsb1v 2999 |
. . . 4
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7 | nfcsb1v 2999 |
. . . 4
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8 | nfcsb1v 2999 |
. . . 4
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9 | 6, 7, 8 | nfbr 3937 |
. . 3
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10 | csbeq1a 2977 |
. . . 4
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11 | csbeq1a 2977 |
. . . 4
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12 | csbeq1a 2977 |
. . . 4
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13 | 10, 11, 12 | breq123d 3907 |
. . 3
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14 | 9, 13 | sbie 1745 |
. 2
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15 | 1, 5, 14 | vtoclbg 2716 |
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 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-sbc 2877 df-csb 2970 df-un 3039 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 |
This theorem is referenced by: sbcbr12g 3943 csbcnvg 4681 sbcfung 5103 csbfv12g 5409 |
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