Theorem preq12i 4273

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 4270	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	mp2an 708	1 |- { A , C } = { B , D }

Syntax hints:   = wceq 1483   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  grpbasex  15994  grpplusgx  15995  indistpsx  20814  lgsdir2lem5  25054  wlk2v2elem2  27016  tgrpset  36033  zlmodzxzadd  42136  zlmodzxzequa  42285  zlmodzxzequap  42288