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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 4874 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 | 
| This theorem is referenced by: sbcop 5493 elxp6 8049 addcompq 10991 mulcompq 10993 addassnq 10999 mulassnq 11000 distrnq 11002 1lt2nq 11014 axi2m1 11200 om2uzrdg 13998 pzriprng1ALT 21508 pzriprng1 21510 precsexlemcbv 28231 axlowdimlem6 28963 clwlkclwwlkflem 30024 konigsbergvtx 30266 konigsbergiedg 30267 nvop2 30628 nvvop 30629 phop 30838 hhsssh 31289 cshw1s2 32946 rngoi 37907 isdrngo1 37964 dfswapf2 48985 swapfcoa 49005 diag1a 49023 | 
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