Theorem tppreq3 4756

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 4741	. . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
2	1	eqcoms 2739	. 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵})
3		tpidm23 4754	. 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
4	2, 3	eqtrdi 2787	1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:   → wi 4   = wceq 1541  {cpr 4624  {ctp 4626

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3949  df-sn 4623  df-pr 4625  df-tp 4627

This theorem is referenced by:  tpprceq3  4800  1to3vfriswmgr  29398