Theorem eqeq12i 2094

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 2093	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	mp2an 416	1 |- ( A = C <-> B = D )

Syntax hints:   <-> wb 103   = wceq 1284

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  rabbi  2531  sbceqg  2922  preqr2g  3559  preqr2  3561  otth  3997  rncoeq  4623  eqfnov  5627  mpt22eqb  5630  f1o2ndf1  5869  ecopovsym  6225  sq11i  9565