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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 4571 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq2d 4097 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵})) |
3 | df-tp 4566 | . 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴}) | |
4 | df-tp 4566 | . 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2803 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∪ cun 3885 {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 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-un 3892 df-sn 4562 df-tp 4566 |
This theorem is referenced by: tpeq3d 4683 tppreq3 4695 fntpb 7085 fztpval 13318 hashtpg 14199 dvh4dimN 39461 |
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