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Theorem tpeq1 3447
 Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq1 (A = B → {A, 𝐶, 𝐷} = {B, 𝐶, 𝐷})

Proof of Theorem tpeq1
StepHypRef Expression
1 preq1 3438 . . 3 (A = B → {A, 𝐶} = {B, 𝐶})
21uneq1d 3090 . 2 (A = B → ({A, 𝐶} ∪ {𝐷}) = ({B, 𝐶} ∪ {𝐷}))
3 df-tp 3375 . 2 {A, 𝐶, 𝐷} = ({A, 𝐶} ∪ {𝐷})
4 df-tp 3375 . 2 {B, 𝐶, 𝐷} = ({B, 𝐶} ∪ {𝐷})
52, 3, 43eqtr4g 2094 1 (A = B → {A, 𝐶, 𝐷} = {B, 𝐶, 𝐷})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∪ cun 2909  {csn 3367  {cpr 3368  {ctp 3369 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-tp 3375 This theorem is referenced by:  tpeq1d  3450
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