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Mirrors > Home > MPE Home > Th. List > opeq12 | Structured version Visualization version 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 4340 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
2 | opeq2 4341 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
3 | 1, 2 | sylan9eq 2664 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: opeq12i 4345 opeq12d 4348 cbvopab 4653 opth 4871 copsex2t 4883 copsex2g 4884 relop 5194 funopg 5836 fvn0ssdmfun 6258 fsn 6308 fnressn 6330 fmptsng 6339 fmptsnd 6340 tpres 6371 cbvoprab12 6627 eqopi 7093 f1o2ndf1 7172 tposoprab 7275 omeu 7552 brecop 7727 ecovcom 7741 ecovass 7742 ecovdi 7743 xpf1o 8007 addsrmo 9773 mulsrmo 9774 addsrpr 9775 mulsrpr 9776 addcnsr 9835 axcnre 9864 seqeq1 12666 opfi1uzind 13138 opfi1uzindOLD 13144 fsumcnv 14346 fprodcnv 14552 eucalgval2 15132 xpstopnlem1 21422 qustgplem 21734 isrusgra 26454 brabgaf 28800 qqhval2 29354 brsegle 31385 finxpreclem3 32406 dvnprodlem1 38836 funop1 40327 |
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