Theorem nfopd

Description: Deduction version of bound-variable hypothesis builder nfop . This shows how the deduction version of a not-free theorem such as nfop can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008)

	Ref	Expression
Hypotheses	nfopd.2	|- ( ph -> F/_ x A )
	nfopd.3	|- ( ph -> F/_ x B )
Assertion	nfopd	|- ( ph -> F/_ x <. A , B >. )

Proof

Step	Hyp	Ref	Expression
1		nfopd.2	|- ( ph -> F/_ x A )
2		nfopd.3	|- ( ph -> F/_ x B )
3		nfaba1	|- F/_ x { z | A. x z e. A }
4		nfaba1	|- F/_ x { z | A. x z e. B }
5	3 4	nfop	|- F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >.
6		nfnfc1	|- F/ x F/_ x A
7		nfnfc1	|- F/ x F/_ x B
8	6 7	nfan	|- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf	|- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	adantr	|- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
11		abidnf	|- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	adantl	|- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
13	10 12	opeq12d	|- ( ( F/_ x A /\ F/_ x B ) -> <. { z | A. x z e. A } , { z | A. x z e. B } >. = <. A , B >. )
14	8 13	nfceqdf	|- ( ( F/_ x A /\ F/_ x B ) -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
15	1 2 14	syl2anc	|- ( ph -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
16	5 15	mpbii	|- ( ph -> F/_ x <. A , B >. )