Theorem eqeq12 2093

Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
eqeq12	|- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Proof of Theorem eqeq12

Step	Hyp	Ref	Expression
1		eqeq1 2087	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2090	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 449	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> 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:  eqeq12i  2094  eqeq12d  2095  eqeqan12d  2096  funopg  4954  tfri3  5976  th3qlem1  6231  xpdom2  6328  xrlttri3  8872  bcn1  9685