Theorem tprot 3815

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 940	. . 3 ⊢ ((x = A ∨ x = B ∨ x = C) ↔ (x = B ∨ x = C ∨ x = A))
2	1	abbii 2465	. 2 ⊢ {x ∣ (x = A ∨ x = B ∨ x = C)} = {x ∣ (x = B ∨ x = C ∨ x = A)}
3		dftp2 3772	. 2 ⊢ {A, B, C} = {x ∣ (x = A ∨ x = B ∨ x = C)}
4		dftp2 3772	. 2 ⊢ {B, C, A} = {x ∣ (x = B ∨ x = C ∨ x = A)}
5	2, 3, 4	3eqtr4i 2383	1 ⊢ {A, B, C} = {B, C, A}

Syntax hints:   ∨ w3o 933   = wceq 1642  {cab 2339  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by:  tpcomb  3817  tpass  3818  tpidm13  3822  tpidm23  3823