Theorem nfopdALT

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, 19-Nov-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfopdALT.1	|- ( ph -> F/_ x A )
	nfopdALT.2	|- ( ph -> F/_ x B )
Assertion	nfopdALT	|- ( ph -> F/_ x <. A , B >. )

Proof

Step	Hyp	Ref	Expression
1		nfopdALT.1	|- ( ph -> F/_ x A )
2		nfopdALT.2	|- ( ph -> F/_ x B )
3		abidnf	|- ( F/_ x A -> { z | A. x z e. A } = A )
4	3	adantr	|- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
5		abidnf	|- ( F/_ x B -> { z | A. x z e. B } = B )
6	5	adantl	|- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
7	4 6	opeq12d	|- ( ( F/_ x A /\ F/_ x B ) -> <. { z | A. x z e. A } , { z | A. x z e. B } >. = <. A , B >. )
8		nfaba1	|- F/_ x { z | A. x z e. A }
9		nfaba1	|- F/_ x { z | A. x z e. B }
10	8 9	nfop	|- F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >.
11	1 2 7 10	nfded2	|- ( ph -> F/_ x <. A , B >. )