Theorem nfifd 3637

Description: Deduction version of nfif 3638. (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 3608	. 2 |- if ( ps , A , B ) = { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) }
2		nfv 1577	. . 3 |- F/ y ph
3		nfifd.3	. . . . . 6 |- ( ph -> F/_ x A )
4	3	nfcrd 2389	. . . . 5 |- ( ph -> F/ x y e. A )
5		nfifd.2	. . . . 5 |- ( ph -> F/ x ps )
6	4, 5	nfand 1617	. . . 4 |- ( ph -> F/ x ( y e. A /\ ps ) )
7		nfifd.4	. . . . . 6 |- ( ph -> F/_ x B )
8	7	nfcrd 2389	. . . . 5 |- ( ph -> F/ x y e. B )
9	5	nfnd 1705	. . . . 5 |- ( ph -> F/ x -. ps )
10	8, 9	nfand 1617	. . . 4 |- ( ph -> F/ x ( y e. B /\ -. ps ) )
11	6, 10	nford 1616	. . 3 |- ( ph -> F/ x ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) )
12	2, 11	nfabd 2395	. 2 |- ( ph -> F/_ x { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) } )
13	1, 12	nfcxfrd 2373	1 |- ( ph -> F/_ x if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716   F/wnf 1509   e. wcel 2202   {cab 2217   F/_wnfc 2362   ifcif 3607

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-if 3608

This theorem is referenced by:  nfif  3638