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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 3549 | . 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 1243 〈cop 3378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: oteq1 3558 oteq2 3559 opth 3974 cbvoprab2 5577 dfplpq2 6452 ltexnqq 6506 nnanq0 6556 addpinq1 6562 prarloclemlo 6592 prarloclem3 6595 prarloclem5 6598 prsrriota 6872 caucvgsrlemfv 6875 caucvgsr 6886 pitonnlem2 6923 pitonn 6924 recidpirq 6934 ax1rid 6951 axrnegex 6953 nntopi 6968 axcaucvglemval 6971 fseq1m1p1 8957 frecuzrdglem 9197 |
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