Theorem preqr2 3771

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 3670	. . 3 |- { C , A } = { A , C }
2		prcom 3670	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2191	. 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 3770	. 2 |- ( { A , C } = { B , C } -> A = B )
7	3, 6	sylbi 121	1 |- ( { C , A } = { C , B } -> A = B )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  preq12b  3772  opth  4239  opthreg  4557