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Mirrors > Home > ILE Home > Th. List > opeq12d | Unicode version |
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
opeq1d.1 |
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opeq12d.2 |
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Ref | Expression |
---|---|
opeq12d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1d.1 |
. 2
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2 | opeq12d.2 |
. 2
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3 | opeq12 3542 |
. 2
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4 | 1, 2, 3 | syl2anc 391 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 |
This theorem is referenced by: nfopd 3557 moop2 3979 fliftfuns 5381 elxp6 5738 dfmpt2 5786 tfrlemi1 5887 qliftfuns 6126 xpassen 6240 xpdom2 6241 dfplpq2 6338 dfmpq2 6339 addpipqqs 6354 mulpipq2 6355 mulpipq 6356 mulpipqqs 6357 mulidnq 6373 addnq0mo 6430 mulnq0mo 6431 addnnnq0 6432 mulnnnq0 6433 nqnq0a 6437 nqnq0m 6438 nq0a0 6440 nq02m 6448 genpdf 6491 genipv 6492 genpelxp 6494 addcomprg 6554 mulcomprg 6556 cauappcvgprlemlim 6633 addsrmo 6671 mulsrmo 6672 addsrpr 6673 mulsrpr 6674 addcnsr 6731 mulcnsr 6732 mulresr 6735 pitonnlem2 6743 pitonn 6744 axaddcom 6754 ax0id 6762 axcnre 6765 frecuzrdgrrn 8875 frec2uzrdg 8876 frecuzrdgsuc 8882 iseqeq1 8894 |
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