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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 3793 | . 2 ⊢ (𝐴 = 𝐶 → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐵〉) | |
2 | opeq2 3794 | . 2 ⊢ (𝐵 = 𝐷 → 〈𝐶, 𝐵〉 = 〈𝐶, 𝐷〉) | |
3 | 1, 2 | sylan9eq 2242 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 〈cop 3610 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 |
This theorem is referenced by: opeq12i 3798 opeq12d 3801 cbvopab 4089 opth 4255 copsex2t 4263 copsex2g 4264 relop 4795 funopg 5269 fsn 5708 fnressn 5722 cbvoprab12 5969 eqopi 6196 f1o2ndf1 6252 tposoprab 6304 brecop 6650 th3q 6665 ecovcom 6667 ecovicom 6668 ecovass 6669 ecoviass 6670 ecovdi 6671 ecovidi 6672 xpf1o 6871 1qec 7416 enq0sym 7460 addnq0mo 7475 mulnq0mo 7476 addnnnq0 7477 mulnnnq0 7478 distrnq0 7487 mulcomnq0 7488 addassnq0 7490 addsrmo 7771 mulsrmo 7772 addsrpr 7773 mulsrpr 7774 axcnre 7909 fsumcnv 11476 fprodcnv 11664 eucalgval2 12084 |
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