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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 4404 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: elxp6 7200 addcompq 9772 mulcompq 9774 addassnq 9780 mulassnq 9781 distrnq 9783 1lt2nq 9795 axi2m1 9980 om2uzrdg 12755 axlowdimlem6 25827 konigsbergvtx 27106 konigsbergiedg 27107 nvop2 27463 nvvop 27464 phop 27673 hhsssh 28126 rngoi 33698 isdrngo1 33755 |
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