Theorem nfifd 4114

Description: Deduction version of nfif 4115. (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		dfif2 4088	. 2 |- if ( ps , A , B ) = { y | ( ( y e. B -> ps ) -> ( y e. A /\ ps ) ) }
2		nfv 1843	. . 3 |- F/ y ph
3		nfifd.4	. . . . . 6 |- ( ph -> F/_ x B )
4	3	nfcrd 2771	. . . . 5 |- ( ph -> F/ x y e. B )
5		nfifd.2	. . . . 5 |- ( ph -> F/ x ps )
6	4, 5	nfimd 1823	. . . 4 |- ( ph -> F/ x ( y e. B -> ps ) )
7		nfifd.3	. . . . . 6 |- ( ph -> F/_ x A )
8	7	nfcrd 2771	. . . . 5 |- ( ph -> F/ x y e. A )
9	8, 5	nfand 1826	. . . 4 |- ( ph -> F/ x ( y e. A /\ ps ) )
10	6, 9	nfimd 1823	. . 3 |- ( ph -> F/ x ( ( y e. B -> ps ) -> ( y e. A /\ ps ) ) )
11	2, 10	nfabd 2785	. 2 |- ( ph -> F/_ x { y | ( ( y e. B -> ps ) -> ( y e. A /\ ps ) ) } )
12	1, 11	nfcxfrd 2763	1 |- ( ph -> F/_ x if ( ps , A , B ) )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   {cab 2608   F/_wnfc 2751   ifcif 4086

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-if 4087

This theorem is referenced by:  nfif  4115  nfxnegd  39668