Theorem aovnuoveq

Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovnuoveq	|- ( (( A F B )) =/= _V -> (( A F B )) = ( A F B ) )

Proof

Step	Hyp	Ref	Expression
1		df-aov	|- (( A F B )) = ( F ''' <. A , B >. )
2	1	neeq1i	|- ( (( A F B )) =/= _V <-> ( F ''' <. A , B >. ) =/= _V )
3		afvnufveq	|- ( ( F ''' <. A , B >. ) =/= _V -> ( F ''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov	|- ( A F B ) = ( F ` <. A , B >. )
5	3 1 4	3eqtr4g	|- ( ( F ''' <. A , B >. ) =/= _V -> (( A F B )) = ( A F B ) )
6	2 5	sylbi	|- ( (( A F B )) =/= _V -> (( A F B )) = ( A F B ) )