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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 4797 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 |
This theorem is referenced by: sbcop 5371 elxp6 7712 addcompq 10360 mulcompq 10362 addassnq 10368 mulassnq 10369 distrnq 10371 1lt2nq 10383 axi2m1 10569 om2uzrdg 13312 axlowdimlem6 26660 clwlkclwwlkflem 27709 konigsbergvtx 27952 konigsbergiedg 27953 nvop2 28312 nvvop 28313 phop 28522 hhsssh 28973 cshw1s2 30561 rngoi 35058 isdrngo1 35115 |
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