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Mirrors > Home > ILE Home > Th. List > tpeq123d | GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Ref | Expression |
---|---|
tpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
tpeq123d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
tpeq123d.3 | ⊢ (𝜑 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
tpeq123d | ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | tpeq1d 3607 | . 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸}) |
3 | tpeq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | tpeq2d 3608 | . 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸}) |
5 | tpeq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | tpeq3d 3609 | . 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹}) |
7 | 2, 4, 6 | 3eqtrd 2174 | 1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 {ctp 3524 |
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 df-tp 3530 |
This theorem is referenced by: fz0tp 9894 fzo0to3tp 9989 |
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