Theorem preq12i 4070

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 4067	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	mp2an 672	1 |- { A , C } = { B , D }

Syntax hints:   = wceq 1370   {cpr 3990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-sn 3989  df-pr 3991

This theorem is referenced by:  grpbasex  14404  grpplusgx  14405  indistpsx  18756  lgsdir2lem5  22809  wlkntrllem2  23638  clwwlkgt0  30605  zlmodzxzadd  30926  zlmodzxzequa  31193  zlmodzxzequap  31196  tgrpset  34752