Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpeq3d Structured version   Visualization version   GIF version

Theorem tpeq3d 4646
 Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
Hypothesis
Ref Expression
tpeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
tpeq3d (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Proof of Theorem tpeq3d
StepHypRef Expression
1 tpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 tpeq3 4643 . 2 (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
31, 2syl 17 1 (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {ctp 4532 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 3444  df-un 3888  df-sn 4529  df-tp 4533 This theorem is referenced by:  tpeq123d  4647  fntpb  6959  erngset  38247  erngset-rN  38255
 Copyright terms: Public domain W3C validator