Theorem feq23 5053

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 5051	. 2 |- ( A = C -> ( F : A --> B <-> F : C --> B ) )
2		feq3 5052	. 2 |- ( B = D -> ( F : C --> B <-> F : C --> D ) )
3	1, 2	sylan9bb 449	1 |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   -->wf 4918

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-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-fn 4925  df-f 4926

This theorem is referenced by:  feq23i  5061