Theorem tppreq3 4716

Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Assertion

Ref	Expression
tppreq3	⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Proof of Theorem tppreq3

Step	Hyp	Ref	Expression
1		tpeq3 4701	. . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
2	1	eqcoms 2768	. 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3		tpidm23 4714	. 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
4	2, 3	eqtrdi 2811	1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1569  {cpr 4586  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4151  df-sn 4585  df-pr 4587  df-tp 4589

This theorem is used by:  tpprceq3  4763  1to3vfriswmgr  30887