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Mirrors > Home > ILE Home > Th. List > opeq12 | GIF version |
Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
opeq12 | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3549 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
2 | opeq2 3550 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
3 | 1, 2 | sylan9eq 2092 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = 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: opeq12i 3554 opeq12d 3557 cbvopab 3828 opth 3974 copsex2t 3982 copsex2g 3983 relop 4486 funopg 4934 fsn 5335 fnressn 5349 cbvoprab12 5578 eqopi 5798 f1o2ndf1 5849 tposoprab 5895 brecop 6196 th3q 6211 ecovcom 6213 ecovicom 6214 ecovass 6215 ecoviass 6216 ecovdi 6217 ecovidi 6218 1qec 6486 enq0sym 6530 addnq0mo 6545 mulnq0mo 6546 addnnnq0 6547 mulnnnq0 6548 distrnq0 6557 mulcomnq0 6558 addassnq0 6560 addsrmo 6828 mulsrmo 6829 addsrpr 6830 mulsrpr 6831 axcnre 6955 |
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