Theorem eqeq12i 2184

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
eqeq12i.1	|- A = B
eqeq12i.2	|- C = D

Assertion

Ref	Expression
eqeq12i	|- ( A = C <-> B = D )

Proof of Theorem eqeq12i

Step	Hyp	Ref	Expression
1		eqeq12i.1	. 2 |- A = B
2		eqeq12i.2	. 2 |- C = D
3		eqeq12 2183	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	mp2an 424	1 |- ( A = C <-> B = D )

Syntax hints:   <-> wb 104   = wceq 1348

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-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  rabbi  2647  sbceqg  3065  preqr2g  3754  preqr2  3756  otth  4227  rncoeq  4884  eqfnov  5959  mpo2eqb  5962  f1o2ndf1  6207  ecopovsym  6609  sq11i  10565  pwle2  14031