![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3693 | . 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸}) |
3 | tpeq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | tpeq2d 3694 | . 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸}) |
5 | tpeq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | tpeq3d 3695 | . 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹}) |
7 | 2, 4, 6 | 3eqtrd 2224 | 1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 {ctp 3606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-sn 3610 df-pr 3611 df-tp 3612 |
This theorem is referenced by: fz0tp 10136 fz0to4untppr 10138 fzo0to3tp 10233 prdsex 12736 imasex 12744 imasival 12745 |
Copyright terms: Public domain | W3C validator |