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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 4340 | . 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 1475 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: axi2m1 9859 s3tpop 13504 strlemor1 15796 2strstr1 15812 2strop1 15814 grpbasex 15819 grpplusgx 15820 mat1dimelbas 20096 mat1dim0 20098 mat1dimid 20099 mat1dimscm 20100 mat1dimmul 20101 indistpsx 20624 uhgrstrrepe 25745 mapfzcons 36297 cusgrsize 40670 |
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