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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 4795 | . 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 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: axi2m1 10569 s3tpop 14259 2strstr1 16593 2strop1 16595 grpbasex 16601 grpplusgx 16602 mat1dimelbas 21008 mat1dim0 21010 mat1dimid 21011 mat1dimscm 21012 mat1dimmul 21013 indistpsx 21546 setsiedg 26748 cusgrsize 27163 mapfzcons 39191 |
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