Theorem nfifd 3532

Description: Deduction version of nfif 3533. (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 3506	. 2 |- if ( ps , A , B ) = { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) }
2		nfv 1508	. . 3 |- F/ y ph
3		nfifd.3	. . . . . 6 |- ( ph -> F/_ x A )
4	3	nfcrd 2313	. . . . 5 |- ( ph -> F/ x y e. A )
5		nfifd.2	. . . . 5 |- ( ph -> F/ x ps )
6	4, 5	nfand 1548	. . . 4 |- ( ph -> F/ x ( y e. A /\ ps ) )
7		nfifd.4	. . . . . 6 |- ( ph -> F/_ x B )
8	7	nfcrd 2313	. . . . 5 |- ( ph -> F/ x y e. B )
9	5	nfnd 1637	. . . . 5 |- ( ph -> F/ x -. ps )
10	8, 9	nfand 1548	. . . 4 |- ( ph -> F/ x ( y e. B /\ -. ps ) )
11	6, 10	nford 1547	. . 3 |- ( ph -> F/ x ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) )
12	2, 11	nfabd 2319	. 2 |- ( ph -> F/_ x { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) } )
13	1, 12	nfcxfrd 2297	1 |- ( ph -> F/_ x if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   F/wnf 1440   e. wcel 2128   {cab 2143   F/_wnfc 2286   ifcif 3505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-if 3506

This theorem is referenced by:  nfif  3533