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Theorem tpeq2 4412
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
tpeq2 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})

Proof of Theorem tpeq2
StepHypRef Expression
1 preq2 4403 . . 3 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
21uneq1d 3915 . 2 (𝐴 = 𝐵 → ({𝐶, 𝐴} ∪ {𝐷}) = ({𝐶, 𝐵} ∪ {𝐷}))
3 df-tp 4319 . 2 {𝐶, 𝐴, 𝐷} = ({𝐶, 𝐴} ∪ {𝐷})
4 df-tp 4319 . 2 {𝐶, 𝐵, 𝐷} = ({𝐶, 𝐵} ∪ {𝐷})
52, 3, 43eqtr4g 2829 1 (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  cun 3719  {csn 4314  {cpr 4316  {ctp 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-sn 4315  df-pr 4317  df-tp 4319
This theorem is referenced by:  tpeq2d  4415  fntpb  6616  fztpval  12608  hashtpg  13468  dvh4dimN  37250  lmod1  42799
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