Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4805 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
2 | opeq2 4806 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
3 | 1, 2 | sylan9eq 2878 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: opeq12i 4810 opeq12d 4813 cbvopab 5139 opth 5370 copsex2t 5385 copsex2g 5386 relop 5723 funopg 6391 fvn0ssdmfun 6844 fsn 6899 fnressn 6922 fmptsng 6932 fmptsnd 6933 tpres 6965 cbvoprab12 7245 eqopi 7727 f1o2ndf1 7820 tposoprab 7930 omeu 8213 brecop 8392 ecovcom 8405 ecovass 8406 ecovdi 8407 xpf1o 8681 addsrmo 10497 mulsrmo 10498 addsrpr 10499 mulsrpr 10500 addcnsr 10559 axcnre 10588 seqeq1 13375 opfi1uzind 13862 fsumcnv 15130 fprodcnv 15339 eucalgval2 15927 xpstopnlem1 22419 qustgplem 22731 finsumvtxdg2size 27334 brabgaf 30361 qqhval2 31225 brsegle 33571 copsex2d 34433 finxpreclem3 34676 eqrelf 35519 dvnprodlem1 42238 or2expropbilem1 43274 or2expropbilem2 43275 funop1 43489 ich2exprop 43640 ichnreuop 43641 ichreuopeq 43642 reuopreuprim 43695 uspgrsprf1 44029 |
Copyright terms: Public domain | W3C validator |