Theorem preq12i 3658

Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Hypotheses

Ref	Expression
preq1i.1	|- A = B
preq12i.2	|- C = D

Assertion

Ref	Expression
preq12i	|- { A , C } = { B , D }

Proof of Theorem preq12i

Step	Hyp	Ref	Expression
1		preq1i.1	. 2 |- A = B
2		preq12i.2	. 2 |- C = D
3		preq12 3655	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	mp2an 423	1 |- { A , C } = { B , D }

Syntax hints:   = wceq 1343   {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:  lgsdir2lem5  13573