Theorem ovmpox2

Description: The value of an operation class abstraction. Variant of ovmpoga which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Mario Carneiro, 20-Dec-2013)

	Ref	Expression
Hypotheses	ovmpox2.1	|- ( ( x = A /\ y = B ) -> R = S )
	ovmpox2.2	|- ( y = B -> C = L )
	ovmpox2.3	|- F = ( x e. C , y e. D |-> R )
Assertion	ovmpox2	|- ( ( A e. L /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Proof

Step	Hyp	Ref	Expression
1		ovmpox2.1	|- ( ( x = A /\ y = B ) -> R = S )
2		ovmpox2.2	|- ( y = B -> C = L )
3		ovmpox2.3	|- F = ( x e. C , y e. D |-> R )
4	3	a1i	|- ( ( A e. L /\ B e. D /\ S e. H ) -> F = ( x e. C , y e. D |-> R ) )
5	1	adantl	|- ( ( ( A e. L /\ B e. D /\ S e. H ) /\ ( x = A /\ y = B ) ) -> R = S )
6	2	adantl	|- ( ( ( A e. L /\ B e. D /\ S e. H ) /\ y = B ) -> C = L )
7		simp1	|- ( ( A e. L /\ B e. D /\ S e. H ) -> A e. L )
8		simp2	|- ( ( A e. L /\ B e. D /\ S e. H ) -> B e. D )
9		simp3	|- ( ( A e. L /\ B e. D /\ S e. H ) -> S e. H )
10	4 5 6 7 8 9	ovmpordx	|- ( ( A e. L /\ B e. D /\ S e. H ) -> ( A F B ) = S )