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| Mirrors > Home > MPE Home > Th. List > tpeq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
| Ref | Expression |
|---|---|
| tpeq1 | ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | preq1 4733 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
| 2 | 1 | uneq1d 4167 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐶} ∪ {𝐷}) = ({𝐵, 𝐶} ∪ {𝐷})) |
| 3 | df-tp 4631 | . 2 ⊢ {𝐴, 𝐶, 𝐷} = ({𝐴, 𝐶} ∪ {𝐷}) | |
| 4 | df-tp 4631 | . 2 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷}) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: tpeq1d 4745 hashtpg 14524 hash3tpde 14532 erngset 40802 erngset-rN 40810 dvh4dimN 41449 cycl3grtri 47914 grimgrtri 47916 grlimgrtri 47963 usgrexmpl1tri 47984 lmod1 48409 |
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