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Mirrors > Home > ILE Home > Th. List > opeq1d | GIF version |
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Ref | Expression |
---|---|
opeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
opeq1d | ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | opeq1 3705 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 〈cop 3530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: oteq1 3714 oteq2 3715 opth 4159 cbvoprab2 5844 djuf1olem 6938 dfplpq2 7162 ltexnqq 7216 nnanq0 7266 addpinq1 7272 prarloclemlo 7302 prarloclem3 7305 prarloclem5 7308 prsrriota 7596 caucvgsrlemfv 7599 caucvgsr 7610 pitonnlem2 7655 pitonn 7656 recidpirq 7666 ax1rid 7685 axrnegex 7687 nntopi 7702 axcaucvglemval 7705 fseq1m1p1 9875 frecuzrdglem 10184 frecuzrdgg 10189 frecuzrdgdomlem 10190 frecuzrdgfunlem 10192 frecuzrdgsuctlem 10196 fsum2dlemstep 11203 ennnfonelemp1 11919 ennnfonelemnn0 11935 |
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