Theorem tprot 4720

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 1108	. . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴))
2	1	abbii 2833	. 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
3		dftp2 4662	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
4		dftp2 4662	. 2 ⊢ {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵 ∨ 𝑥 = 𝐶 ∨ 𝑥 = 𝐴)}
5	2, 3, 4	3eqtr4i 2799	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

This proof depends on syntax axioms:   ∨ w3o 1102   = wceq 1570  {cab 2744  {ctp 4598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-un 3913  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is used by:  tpcomb  4722  tpass  4723  tpidm13  4727  tpidm23  4728  tpprceq3  4777  fvtp2  7201  fvtp3  7202  fvtp2g  7204  fvtp3g  7205  f13dfv  7283  en3lplem2  9592  estrres  18220  ex-chn2  18719  nb3grprlem2  29768  nb3grpr  29769  nb3grpr2  29770  nb3gr2nb  29771  cplgr3v  29822  frgr3v  30663  1to3vfriswmgr  30668  tpssbd  32923  tpsscd  32924  dvh4dimN  42262  en3lplem2VD  45593