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Mirrors > Home > ILE Home > Th. List > opelopabsbALT | Unicode version |
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4272, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opelopabsbALT |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 1674 |
. . 3
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2 | vex 2752 |
. . . . . . 7
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3 | vex 2752 |
. . . . . . 7
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4 | 2, 3 | opth 4249 |
. . . . . 6
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5 | equcom 1716 |
. . . . . . 7
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6 | equcom 1716 |
. . . . . . 7
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7 | 5, 6 | anbi12ci 461 |
. . . . . 6
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8 | 4, 7 | bitri 184 |
. . . . 5
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9 | 8 | anbi1i 458 |
. . . 4
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10 | 9 | 2exbii 1616 |
. . 3
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11 | 1, 10 | bitri 184 |
. 2
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12 | elopab 4270 |
. 2
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13 | 2sb5 1993 |
. 2
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14 | 11, 12, 13 | 3bitr4i 212 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 |
This theorem is referenced by: inopab 4771 cnvopab 5042 |
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