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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 4663 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
2 | 1 | uneq1d 4138 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐶} ∪ {𝐷}) = ({𝐵, 𝐶} ∪ {𝐷})) |
3 | df-tp 4566 | . 2 ⊢ {𝐴, 𝐶, 𝐷} = ({𝐴, 𝐶} ∪ {𝐷}) | |
4 | df-tp 4566 | . 2 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷}) | |
5 | 2, 3, 4 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∪ cun 3934 {csn 4561 {cpr 4563 {ctp 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3941 df-sn 4562 df-pr 4564 df-tp 4566 |
This theorem is referenced by: tpeq1d 4675 hashtpg 13837 erngset 37930 erngset-rN 37938 dvh4dimN 38577 lmod1 44540 |
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