Theorem tprot 4774

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 1092	. . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴))
2	1	abbii 2812	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
3		dftp2 4714	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
4		dftp2 4714	. 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
5	2, 3, 4	3eqtr4i 2778	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

Syntax hints:   ∨ w3o 1086   = wceq 1537  {cab 2717  {ctp 4652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651  df-tp 4653

This theorem is referenced by:  tpcomb  4776  tpass  4777  tpidm13  4781  tpidm23  4782  tpprceq3  4829  fvtp2  7233  fvtp3  7234  fvtp2g  7236  fvtp3g  7237  f13dfv  7310  en3lplem2  9682  estrres  18208  nb3grprlem2  29416  nb3grpr  29417  nb3grpr2  29418  nb3gr2nb  29419  cplgr3v  29470  frgr3v  30307  1to3vfriswmgr  30312  dvh4dimN  41404  en3lplem2VD  44815