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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 4877 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
2 | opeq2 4878 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
3 | 1, 2 | sylan9eq 2794 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 〈cop 4636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: opeq12i 4882 opeq12d 4885 cbvopab 5219 cbvopabv 5220 opth 5486 copsex2t 5502 relop 5863 funopg 6601 fvn0ssdmfun 7093 fsn 7154 fnressn 7177 fmptsng 7187 fmptsnd 7188 tpres 7220 cbvoprab12 7521 cbvoprab12v 7522 eqopi 8048 f1o2ndf1 8145 tposoprab 8285 omeu 8621 brecop 8848 ecovcom 8861 ecovass 8862 ecovdi 8863 xpf1o 9177 addsrmo 11110 mulsrmo 11111 addsrpr 11112 mulsrpr 11113 addcnsr 11172 axcnre 11201 seqeq1 14041 opfi1uzind 14546 fsumcnv 15805 fprodcnv 16015 eucalgval2 16614 xpstopnlem1 23832 qustgplem 24144 finsumvtxdg2size 29582 brabgaf 32627 qqhval2 33944 brsegle 36089 copsex2d 37121 finxpreclem3 37375 eqrelf 38236 dvnprodlem1 45901 or2expropbilem1 46981 or2expropbilem2 46982 funop1 47232 ich2exprop 47395 ichnreuop 47396 ichreuopeq 47397 reuopreuprim 47450 uspgrsprf1 47990 |
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