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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 3713 | . 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 1332 〈cop 3535 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: oteq1 3722 oteq2 3723 opth 4167 cbvoprab2 5852 djuf1olem 6946 dfplpq2 7186 ltexnqq 7240 nnanq0 7290 addpinq1 7296 prarloclemlo 7326 prarloclem3 7329 prarloclem5 7332 prsrriota 7620 caucvgsrlemfv 7623 caucvgsr 7634 pitonnlem2 7679 pitonn 7680 recidpirq 7690 ax1rid 7709 axrnegex 7711 nntopi 7726 axcaucvglemval 7729 fseq1m1p1 9906 frecuzrdglem 10215 frecuzrdgg 10220 frecuzrdgdomlem 10221 frecuzrdgfunlem 10223 frecuzrdgsuctlem 10227 fsum2dlemstep 11235 ennnfonelemp1 11955 ennnfonelemnn0 11971 |
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