Theorem preqr2g 3747

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 3749. (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 3652	. . 3 |- { C , A } = { A , C }
2		prcom 3652	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2179	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preqr1g 3746	. 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 1343   e. wcel 2136   _Vcvv 2726   {cpr 3577

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  opth  4215