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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 4434 | . 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 1523 〈cop 4216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 |
This theorem is referenced by: fnressn 6465 fressnfv 6467 wfrlem14 7473 seqomlem1 7590 recmulnq 9824 addresr 9997 seqval 12852 ids1 13413 wrdeqs1cat 13520 swrdccat3a 13540 ressinbas 15983 oduval 17177 mgmnsgrpex 17465 sgrpnmndex 17466 efgi0 18179 efgi1 18180 vrgpinv 18228 frgpnabllem1 18322 mat1dimid 20328 uspgr1v1eop 26186 clwlksfoclwwlk 27050 wlk2v2e 27135 avril1 27449 nvop 27659 phop 27801 signstfveq0 30782 bnj601 31116 tgrpset 36350 erngset 36405 erngset-rN 36413 pfx1 41736 pfxccatpfx2 41753 zlmodzxzadd 42461 lmod1 42606 lmod1zr 42607 zlmodzxzequa 42610 zlmodzxzequap 42613 |
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