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Mirrors > Home > MPE Home > Th. List > tpeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
tpeq3 | ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4535 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq2d 4090 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵})) |
3 | df-tp 4530 | . 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴}) | |
4 | df-tp 4530 | . 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2858 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∪ cun 3879 {csn 4525 {cpr 4527 {ctp 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-tp 4530 |
This theorem is referenced by: tpeq3d 4643 tppreq3 4655 fntpb 6949 fztpval 12964 hashtpg 13839 dvh4dimN 38743 |
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