Theorem preq12i 3671

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 3668	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	mp2an 426	1 |- { A , C } = { B , D }

Syntax hints:   = wceq 1353   {cpr 3590

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596

This theorem is referenced by:  lgsdir2lem5  14002