Theorem feq23 5258

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 5256	. 2 |- ( A = C -> ( F : A --> B <-> F : C --> B ) )
2		feq3 5257	. 2 |- ( B = D -> ( F : C --> B <-> F : C --> D ) )
3	1, 2	sylan9bb 457	1 |- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   -->wf 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-fn 5126  df-f 5127

This theorem is referenced by:  feq23i  5267