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| Mirrors > Home > MPE Home > Th. List > opeq1i | 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 |
|---|---|
| opeq1i | ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | opeq1 4873 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: axi2m1 11199 s3tpop 14948 2strstr1OLD 17271 2strop1 17273 grpbasex 17335 grpplusgx 17336 mat1dimelbas 22477 mat1dim0 22479 mat1dimid 22480 mat1dimscm 22481 mat1dimmul 22482 indistpsx 23017 nosupcbv 27747 noinfcbv 27762 setsiedg 29053 cusgrsize 29472 mapfzcons 42727 |
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