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Mirrors > Home > ILE Home > Th. List > tpeq3 | GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
tpeq3 | ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3538 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq2d 3230 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵})) |
3 | df-tp 3535 | . 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴}) | |
4 | df-tp 3535 | . 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∪ cun 3069 {csn 3527 {cpr 3528 {ctp 3529 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-tp 3535 |
This theorem is referenced by: tpeq3d 3614 tppreq3 3626 fztpval 9863 |
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