Theorem preqr2g 3689

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 3691. (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 3594	. . 3 |- { C , A } = { A , C }
2		prcom 3594	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2151	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preqr1g 3688	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )
5	3, 4	syl5bi 151	1 |- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2681   {cpr 3523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  opth  4154