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Mirrors > Home > NFE Home > Th. List > preqr1 | Unicode version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
preqr1.1 | |
preqr1.2 |
Ref | Expression |
---|---|
preqr1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preqr1.1 | . . . . 5 | |
2 | 1 | prid1 3827 | . . . 4 |
3 | eleq2 2414 | . . . 4 | |
4 | 2, 3 | mpbii 202 | . . 3 |
5 | 1 | elpr 3751 | . . 3 |
6 | 4, 5 | sylib 188 | . 2 |
7 | preqr1.2 | . . . . 5 | |
8 | 7 | prid1 3827 | . . . 4 |
9 | eleq2 2414 | . . . 4 | |
10 | 8, 9 | mpbiri 224 | . . 3 |
11 | 7 | elpr 3751 | . . 3 |
12 | 10, 11 | sylib 188 | . 2 |
13 | eqcom 2355 | . 2 | |
14 | eqeq2 2362 | . 2 | |
15 | 6, 12, 13, 14 | oplem1 930 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 357 wceq 1642 wcel 1710 cvv 2859 cpr 3738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 |
This theorem is referenced by: preqr2 4125 |
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