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Mirrors > Home > MPE Home > Th. List > opeq12i | Structured version Visualization version GIF version |
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
Ref | Expression |
---|---|
opeq1i.1 | ⊢ 𝐴 = 𝐵 |
opeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
opeq12i | ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | opeq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | opeq12 4541 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
4 | 1, 2, 3 | mp2an 672 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 〈cop 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 |
This theorem is referenced by: elxp6 7349 addcompq 9974 mulcompq 9976 addassnq 9982 mulassnq 9983 distrnq 9985 1lt2nq 9997 axi2m1 10182 om2uzrdg 12959 axlowdimlem6 26044 clwlkclwwlkflem 27150 konigsbergvtx 27422 konigsbergiedg 27423 nvop2 27799 nvvop 27800 phop 28009 hhsssh 28462 rngoi 34026 isdrngo1 34083 |
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