Theorem preqr2g 3890

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3892. (Contributed by Jim Kingdon, 21-Sep-2018.)

Assertion

Ref	Expression
preqr2g	|- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )

Proof of Theorem preqr2g

Step	Hyp	Ref	Expression
1		prcom 3786	. . 3 |- { C , A } = { A , C }
2		prcom 3786	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2252	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preqr1g 3889	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )
5	3, 4	biimtrid 152	1 |- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  opth  4375