Theorem ovmpodv

Description: Alternate deduction version of ovmpo , suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017)

	Ref	Expression
Hypotheses	ovmpodf.1	|- ( ph -> A e. C )
	ovmpodf.2	|- ( ( ph /\ x = A ) -> B e. D )
	ovmpodf.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
	ovmpodf.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
Assertion	ovmpodv	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ovmpodf.1	|- ( ph -> A e. C )
2		ovmpodf.2	|- ( ( ph /\ x = A ) -> B e. D )
3		ovmpodf.3	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R e. V )
4		ovmpodf.4	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ( A F B ) = R -> ps ) )
5		nfcv	|- F/_ x F
6		nfv	|- F/ x ps
7		nfcv	|- F/_ y F
8		nfv	|- F/ y ps
9	1 2 3 4 5 6 7 8	ovmpodf	|- ( ph -> ( F = ( x e. C , y e. D |-> R ) -> ps ) )