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| Mirrors > Home > MPE Home > Th. List > opeq2i | Structured version Visualization version GIF version | ||
| Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
| Ref | Expression |
|---|---|
| opeq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| opeq2i | ⊢ 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | opeq2 4874 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: fnressn 7178 fressnfv 7180 wfrlem14OLD 8362 seqomlem1 8490 recmulnq 11004 addresr 11178 seqval 14053 ids1 14635 pfx1 14741 pfxccatpfx2 14775 ressinbas 17291 oduval 18333 mgmnsgrpex 18944 sgrpnmndex 18945 efgi0 19738 efgi1 19739 vrgpinv 19787 frgpnabllem1 19891 dfcnfldOLD 21380 pzriprng1ALT 21507 mat1dimid 22480 seqsval 28294 uspgr1v1eop 29266 wlk2v2e 30176 avril1 30482 nvop 30695 phop 30837 bnj601 34934 tgrpset 40747 erngset 40802 erngset-rN 40810 stgr0 47927 stgr1 47928 zlmodzxzadd 48274 lmod1 48409 lmod1zr 48410 zlmodzxzequa 48413 zlmodzxzequap 48416 |
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