Theorem eqeq12i 2171

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 2170	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	mp2an 423	1 |- ( A = C <-> B = D )

Syntax hints:   <-> wb 104   = wceq 1335

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150

This theorem is referenced by:  rabbi  2634  sbceqg  3047  preqr2g  3731  preqr2  3733  otth  4203  rncoeq  4860  eqfnov  5928  mpo2eqb  5931  f1o2ndf1  6176  ecopovsym  6577  sq11i  10512  pwle2  13612