Theorem nfopdALT 34258

Description: Deduction version of bound-variable hypothesis builder nfop 4418. This shows how the deduction version of a not-free theorem such as nfop 4418 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem nfopdALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfopdALT.1	. 2 |- ( ph -> F/_ x A )
2		nfopdALT.2	. 2 |- ( ph -> F/_ x B )
3		abidnf 3375	. . . 4 |- ( F/_ x A -> { z | A. x z e. A } = A )
4	3	adantr 481	. . 3 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
5		abidnf 3375	. . . 4 |- ( F/_ x B -> { z | A. x z e. B } = B )
6	5	adantl 482	. . 3 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
7	4, 6	opeq12d 4410	. 2 |- ( ( F/_ x A /\ F/_ x B ) -> <. { z | A. x z e. A } , { z | A. x z e. B } >. = <. A , B >. )
8		nfaba1 2770	. . 3 |- F/_ x { z | A. x z e. A }
9		nfaba1 2770	. . 3 |- F/_ x { z | A. x z e. B }
10	8, 9	nfop 4418	. 2 |- F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >.
11	1, 2, 7, 10	nfded2 34255	1 |- ( ph -> F/_ x <. A , B >. )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by: (None)