Theorem tpeq3 4725

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
tpeq3	⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Proof of Theorem tpeq3

Step	Hyp	Ref	Expression
1		sneq 4616	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2	1	uneq2d 4148	. 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵}))
3		df-tp 4611	. 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴})
4		df-tp 4611	. 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵})
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3929  {csn 4606  {cpr 4608  {ctp 4610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-tp 4611

This theorem is referenced by:  tpeq3d  4728  tppreq3  4740  fntpb  7206  fztpval  13608  hashtpg  14508  dvh4dimN  41471  cycl3grtri  47926  grimgrtri  47928  usgrgrtrirex  47929  grlimgrtri  47975  usgrexmpl1tri  47996