Theorem feq23 5459

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 5457	. 2 |- ( A = C -> ( F : A --> B <-> F : C --> B ) )
2		feq3 5458	. 2 |- ( B = D -> ( F : C --> B <-> F : C --> D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   -->wf 5314

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-fn 5321  df-f 5322

This theorem is referenced by:  feq23i  5468