Theorem preq12i 3715

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 3712	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	mp2an 426	1 |- { A , C } = { B , D }

Syntax hints:   = wceq 1373   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  lgsdir2lem5  15509