Theorem eqeq12i 2210

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 2209	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	mp2an 426	1 |- ( A = C <-> B = D )

Syntax hints:   <-> wb 105   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  rabbi  2675  sbceqg  3100  preqr2g  3797  preqr2  3799  otth  4275  rncoeq  4939  eqfnov  6029  mpo2eqb  6032  f1o2ndf1  6286  ecopovsym  6690  sq11i  10721  dvmptfsum  14961  pwle2  15643