Theorem ndmovcom

Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995)

	Ref	Expression
Hypothesis	ndmov.1	|- dom F = ( S X. S )
Assertion	ndmovcom	|- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = ( B F A ) )

Proof

Step	Hyp	Ref	Expression
1		ndmov.1	|- dom F = ( S X. S )
2	1	ndmov	|- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = (/) )
3		ancom	|- ( ( A e. S /\ B e. S ) <-> ( B e. S /\ A e. S ) )
4	1	ndmov	|- ( -. ( B e. S /\ A e. S ) -> ( B F A ) = (/) )
5	3 4	sylnbi	|- ( -. ( A e. S /\ B e. S ) -> ( B F A ) = (/) )
6	2 5	eqtr4d	|- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = ( B F A ) )