Theorem nfifd 3685

Description: Deduction version of nfif 3686. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
nfifd.2	|- (ph -> F/xps)
nfifd.3	|- (ph -> F/_xA)
nfifd.4	|- (ph -> F/_xB)

Assertion

Ref	Expression
nfifd	|- (ph -> F/_xif(ps, A, B))

Proof of Theorem nfifd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif2 3664	. 2 |- if(ps, A, B) = {y | ((y e. B -> ps) -> (y e. A /\ ps))}
2		nfv 1619	. . 3 |- F/yph
3		nfifd.4	. . . . . 6 |- (ph -> F/_xB)
4	3	nfcrd 2502	. . . . 5 |- (ph -> F/x y e. B)
5		nfifd.2	. . . . 5 |- (ph -> F/xps)
6	4, 5	nfimd 1808	. . . 4 |- (ph -> F/x(y e. B -> ps))
7		nfifd.3	. . . . . 6 |- (ph -> F/_xA)
8	7	nfcrd 2502	. . . . 5 |- (ph -> F/x y e. A)
9	8, 5	nfand 1822	. . . 4 |- (ph -> F/x(y e. A /\ ps))
10	6, 9	nfimd 1808	. . 3 |- (ph -> F/x((y e. B -> ps) -> (y e. A /\ ps)))
11	2, 10	nfabd 2508	. 2 |- (ph -> F/_x{y | ((y e. B -> ps) -> (y e. A /\ ps))})
12	1, 11	nfcxfrd 2487	1 |- (ph -> F/_xif(ps, A, B))

Syntax hints:   -> wi 4   /\ wa 358   F/wnf 1544   e. wcel 1710  {cab 2339   F/_wnfc 2476  ifcif 3662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-if 3663

This theorem is referenced by:  nfif  3686