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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 4763 | . 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 1538 〈cop 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: axi2m1 10570 s3tpop 14262 2strstr1 16597 2strop1 16599 grpbasex 16605 grpplusgx 16606 mat1dimelbas 21076 mat1dim0 21078 mat1dimid 21079 mat1dimscm 21080 mat1dimmul 21081 indistpsx 21615 setsiedg 26829 cusgrsize 27244 mapfzcons 39657 |
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