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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 4873 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
| 2 | opeq2 4874 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
| 3 | 1, 2 | sylan9eq 2797 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: opeq12i 4878 opeq12d 4881 cbvopab 5215 cbvopabv 5216 opth 5481 copsex2t 5497 relop 5861 funopg 6600 fvn0ssdmfun 7094 fsn 7155 fnressn 7178 fmptsng 7188 fmptsnd 7189 tpres 7221 cbvoprab12 7522 cbvoprab12v 7523 eqopi 8050 f1o2ndf1 8147 tposoprab 8287 omeu 8623 brecop 8850 ecovcom 8863 ecovass 8864 ecovdi 8865 xpf1o 9179 addsrmo 11113 mulsrmo 11114 addsrpr 11115 mulsrpr 11116 addcnsr 11175 axcnre 11204 seqeq1 14045 opfi1uzind 14550 fsumcnv 15809 fprodcnv 16019 eucalgval2 16618 xpstopnlem1 23817 qustgplem 24129 finsumvtxdg2size 29568 brabgaf 32620 qqhval2 33983 brsegle 36109 copsex2d 37140 finxpreclem3 37394 eqrelf 38256 dvnprodlem1 45961 or2expropbilem1 47044 or2expropbilem2 47045 funop1 47295 ich2exprop 47458 ichnreuop 47459 ichreuopeq 47460 reuopreuprim 47513 uspgrsprf1 48063 |
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