Theorem tppreq3 4717

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 4702	. . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
2	1	eqcoms 2769	. 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3		tpidm23 4715	. 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
4	2, 3	eqtrdi 2812	1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:   → wi 4   = wceq 1559  {cpr 4583  {ctp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584  df-tp 4586

This theorem is referenced by:  tpprceq3  4763  1to3vfriswmgr  30428