Theorem tprot 4713

Description: Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
tprot	⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

Proof of Theorem tprot

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3orrot 1091	. . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴))
2	1	abbii 2796	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
3		dftp2 4655	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
4		dftp2 4655	. 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
5	2, 3, 4	3eqtr4i 2762	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

Syntax hints:   ∨ w3o 1085   = wceq 1540  {cab 2707  {ctp 4593

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592  df-tp 4594

This theorem is referenced by:  tpcomb  4715  tpass  4716  tpidm13  4720  tpidm23  4721  tpprceq3  4768  fvtp2  7170  fvtp3  7171  fvtp2g  7173  fvtp3g  7174  f13dfv  7249  en3lplem2  9566  estrres  18100  nb3grprlem2  29308  nb3grpr  29309  nb3grpr2  29310  nb3gr2nb  29311  cplgr3v  29362  frgr3v  30204  1to3vfriswmgr  30209  tpssbd  32469  tpsscd  32470  dvh4dimN  41441  en3lplem2VD  44833