Theorem preqr2 3872

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
preqr2.1	|- A e. _V
preqr2.2	|- B e. _V

Assertion

Ref	Expression
preqr2	|- ( { C , A } = { C , B } -> A = B )

Proof of Theorem preqr2

Step	Hyp	Ref	Expression
1		prcom 3766	. . 3 |- { C , A } = { A , C }
2		prcom 3766	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2246	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preqr2.1	. . 3 |- A e. _V
5		preqr2.2	. . 3 |- B e. _V
6	4, 5	preqr1 3871	. 2 |- ( { A , C } = { B , C } -> A = B )
7	3, 6	sylbi 121	1 |- ( { C , A } = { C , B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2812   {cpr 3689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695

This theorem is referenced by:  preq12b  3873  opth  4352  opthreg  4677  usgredgreu  16198  uspgredg2vtxeu  16200