Theorem preqr2g 3580

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 3582. (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 3487	. . 3 |- { C , A } = { A , C }
2		prcom 3487	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2096	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preqr1g 3579	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )
5	3, 4	syl5bi 150	1 |- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2610   {cpr 3418

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-sn 3423  df-pr 3424

This theorem is referenced by:  opth  4021