Theorem nfifd 3355

Description: Deduction version of nfif 3356. (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 3332	. 2 |- if ( ps , A , B ) = { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) }
2		nfv 1421	. . 3 |- F/ y ph
3		nfifd.3	. . . . . 6 |- ( ph -> F/_ x A )
4	3	nfcrd 2191	. . . . 5 |- ( ph -> F/ x y e. A )
5		nfifd.2	. . . . 5 |- ( ph -> F/ x ps )
6	4, 5	nfand 1460	. . . 4 |- ( ph -> F/ x ( y e. A /\ ps ) )
7		nfifd.4	. . . . . 6 |- ( ph -> F/_ x B )
8	7	nfcrd 2191	. . . . 5 |- ( ph -> F/ x y e. B )
9	5	nfnd 1547	. . . . 5 |- ( ph -> F/ x -. ps )
10	8, 9	nfand 1460	. . . 4 |- ( ph -> F/ x ( y e. B /\ -. ps ) )
11	6, 10	nford 1459	. . 3 |- ( ph -> F/ x ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) )
12	2, 11	nfabd 2196	. 2 |- ( ph -> F/_ x { y | ( ( y e. A /\ ps ) \/ ( y e. B /\ -. ps ) ) } )
13	1, 12	nfcxfrd 2176	1 |- ( ph -> F/_ x if ( ps , A , B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ wo 629   F/wnf 1349   e. wcel 1393   {cab 2026   F/_wnfc 2165   ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-if 3332

This theorem is referenced by:  nfif  3356