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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 4699 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
2 | 1 | uneq1d 4127 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴, 𝐶} ∪ {𝐷}) = ({𝐵, 𝐶} ∪ {𝐷})) |
3 | df-tp 4596 | . 2 ⊢ {𝐴, 𝐶, 𝐷} = ({𝐴, 𝐶} ∪ {𝐷}) | |
4 | df-tp 4596 | . 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 1542 ∪ cun 3913 {csn 4591 {cpr 4593 {ctp 4595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 df-tp 4596 |
This theorem is referenced by: tpeq1d 4711 hashtpg 14391 erngset 39292 erngset-rN 39300 dvh4dimN 39939 lmod1 46647 |
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