Theorem eqeq12 2190

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 2184	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2187	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  eqeq12i  2191  eqeq12d  2192  eqeqan12d  2193  funopg  5252  tfri3  6370  th3qlem1  6639  xpdom2  6833  difinfsnlem  7100  difinfsn  7101  xrlttri3  9799  bcn1  10740  summodc  11393  prodmodc  11588  ringinvnz1ne0  13231