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Mirrors > Home > MPE Home > Th. List > elpreqpr | Structured version Visualization version Unicode version |
Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.) |
Ref | Expression |
---|---|
elpreqpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3040 |
. 2
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2 | elpri 3976 |
. 2
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3 | elpreqprlem 4152 |
. . . . 5
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4 | eleq1 2537 |
. . . . . 6
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5 | preq1 4042 |
. . . . . . . 8
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6 | 5 | eqeq2d 2481 |
. . . . . . 7
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7 | 6 | exbidv 1776 |
. . . . . 6
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8 | 4, 7 | imbi12d 327 |
. . . . 5
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9 | 3, 8 | mpbiri 241 |
. . . 4
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10 | 9 | impcom 437 |
. . 3
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11 | elpreqprlem 4152 |
. . . . . 6
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12 | prcom 4041 |
. . . . . . . 8
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13 | 12 | eqeq1i 2476 |
. . . . . . 7
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14 | 13 | exbii 1726 |
. . . . . 6
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15 | 11, 14 | sylib 201 |
. . . . 5
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16 | eleq1 2537 |
. . . . . 6
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17 | preq1 4042 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | eqeq2d 2481 |
. . . . . . 7
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19 | 18 | exbidv 1776 |
. . . . . 6
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20 | 16, 19 | imbi12d 327 |
. . . . 5
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21 | 15, 20 | mpbiri 241 |
. . . 4
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22 | 21 | impcom 437 |
. . 3
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23 | 10, 22 | jaodan 802 |
. 2
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24 | 1, 2, 23 | syl2anc 673 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-v 3033 df-dif 3393 df-un 3395 df-nul 3723 df-sn 3960 df-pr 3962 |
This theorem is referenced by: elpreqprb 4154 |
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