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Mirrors > Home > ILE Home > Th. List > tpeq2 | GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
tpeq2 | ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 3609 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | |
2 | 1 | uneq1d 3234 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷})) |
3 | df-tp 3540 | . 2 ⊢ {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷}) | |
4 | df-tp 3540 | . 2 ⊢ {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷}) | |
5 | 2, 3, 4 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∪ cun 3074 {csn 3532 {cpr 3533 {ctp 3534 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-tp 3540 |
This theorem is referenced by: tpeq2d 3621 fztpval 9894 |
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