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Mirrors > Home > ILE Home > Th. List > opeq12 | 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 3779 | . 2 ⊢ (𝐴 = 𝐶 → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩) | |
2 | opeq2 3780 | . 2 ⊢ (𝐵 = 𝐷 → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩) | |
3 | 1, 2 | sylan9eq 2230 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ⟨cop 3596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2740 df-un 3134 df-sn 3599 df-pr 3600 df-op 3602 |
This theorem is referenced by: opeq12i 3784 opeq12d 3787 cbvopab 4075 opth 4238 copsex2t 4246 copsex2g 4247 relop 4778 funopg 5251 fsn 5689 fnressn 5703 cbvoprab12 5949 eqopi 6173 f1o2ndf1 6229 tposoprab 6281 brecop 6625 th3q 6640 ecovcom 6642 ecovicom 6643 ecovass 6644 ecoviass 6645 ecovdi 6646 ecovidi 6647 xpf1o 6844 1qec 7387 enq0sym 7431 addnq0mo 7446 mulnq0mo 7447 addnnnq0 7448 mulnnnq0 7449 distrnq0 7458 mulcomnq0 7459 addassnq0 7461 addsrmo 7742 mulsrmo 7743 addsrpr 7744 mulsrpr 7745 axcnre 7880 fsumcnv 11445 fprodcnv 11633 eucalgval2 12053 |
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