Theorem nfifd 3597

Description: Deduction version of nfif 3598. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
nfifd.2	|- ( ph -> F/ x ps )
nfifd.3	|- ( ph -> F/_ x A )
nfifd.4	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfifd	|- ( ph -> F/_ x if ( ps , A , B ) )

Proof of Theorem nfifd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-if 3571	. 2 |- if ( ps , A , B ) = { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) }
2		nfv 1550	. . 3 |- F/ y ph
3		nfifd.3	. . . . . 6 |- ( ph -> F/_ x A )
4	3	nfcrd 2361	. . . . 5 |- ( ph -> F/ x y e. A )
5		nfifd.2	. . . . 5 |- ( ph -> F/ x ps )
6	4, 5	nfand 1590	. . . 4 |- ( ph -> F/ x ( y e. A /\ ps ) )
7		nfifd.4	. . . . . 6 |- ( ph -> F/_ x B )
8	7	nfcrd 2361	. . . . 5 |- ( ph -> F/ x y e. B )
9	5	nfnd 1679	. . . . 5 |- ( ph -> F/ x -. ps )
10	8, 9	nfand 1590	. . . 4 |- ( ph -> F/ x ( y e. B /\ -. ps ) )
11	6, 10	nford 1589	. . 3 |- ( ph -> F/ x ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) )
12	2, 11	nfabd 2367	. 2 |- ( ph -> F/_ x { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) } )
13	1, 12	nfcxfrd 2345	1 |- ( ph -> F/_ x if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   F/wnf 1482   e. wcel 2175   {cab 2190   F/_wnfc 2334   ifcif 3570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-if 3571

This theorem is referenced by:  nfif  3598