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Mirrors > Home > MPE Home > Th. List > sbcel2 | Structured version Visualization version Unicode version |
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcel2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcel12 3784 |
. . 3
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2 | csbconstg 3388 |
. . . 4
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3 | 2 | eleq1d 2524 |
. . 3
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4 | 1, 3 | syl5bb 265 |
. 2
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5 | sbcex 3289 |
. . . 4
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6 | 5 | con3i 142 |
. . 3
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7 | noel 3747 |
. . . 4
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8 | csbprc 3782 |
. . . . 5
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9 | 8 | eleq2d 2525 |
. . . 4
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10 | 7, 9 | mtbiri 309 |
. . 3
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11 | 6, 10 | 2falsed 357 |
. 2
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12 | 4, 11 | pm2.61i 169 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-tru 1458 df-fal 1461 df-ex 1675 df-nf 1679 df-sb 1809 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-in 3423 df-ss 3430 df-nul 3744 |
This theorem is referenced by: csbcom 3795 sbccsb 3805 sbnfc2 3808 csbab 3809 sbcssg 3892 csbuni 4240 csbxp 4935 csbdm 5048 issubc 15789 nbgraopALT 25201 esum2dlem 28962 bj-sbeq 31548 bj-sbceqgALT 31549 bj-sels 31601 f1omptsnlem 31783 csbcom2fi 32414 disjinfi 37506 iccelpart 38785 |
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