Theorem preqr2 3852

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 3747	. . 3 |- { C , A } = { A , C }
2		prcom 3747	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2245	. 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 3851	. 2 |- ( { A , C } = { B , C } -> A = B )
7	3, 6	sylbi 121	1 |- ( { C , A } = { C , B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  preq12b  3853  opth  4329  opthreg  4654  usgredgreu  16066  uspgredg2vtxeu  16068