Theorem feq23 5194

Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
feq23	|- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )

Proof of Theorem feq23

Step	Hyp	Ref	Expression
1		feq2 5192	. 2 |- ( A = C -> ( F : A --> B <-> F : C --> B ) )
2		feq3 5193	. 2 |- ( B = D -> ( F : C --> B <-> F : C --> D ) )
3	1, 2	sylan9bb 453	1 |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   -->wf 5055

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034  df-fn 5062  df-f 5063

This theorem is referenced by:  feq23i  5203