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Mirrors > Home > ILE Home > Th. List > tpeq3d | GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Ref | Expression |
---|---|
tpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
tpeq3d | ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | tpeq3 3458 | . 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 {ctp 3377 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-tp 3383 |
This theorem is referenced by: tpeq123d 3462 |
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