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Mirrors > Home > ILE Home > Th. List > preq12 | GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq12 | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3595 | . 2 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
2 | preq2 3596 | . 2 ⊢ (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷}) | |
3 | 1, 2 | sylan9eq 2190 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 {cpr 3523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 |
This theorem is referenced by: preq12i 3600 preq12d 3603 preq12b 3692 opthreg 4466 relop 4684 qtopbasss 12679 |
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