Theorem eqeq12i 2179

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 2178	. 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 1343

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  rabbi  2643  sbceqg  3061  preqr2g  3747  preqr2  3749  otth  4220  rncoeq  4877  eqfnov  5948  mpo2eqb  5951  f1o2ndf1  6196  ecopovsym  6597  sq11i  10544  pwle2  13878