Theorem dfif2 3665

Description: An alternate definition of the conditional operator df-if 3664 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)

Assertion

Ref	Expression
dfif2	|- if(ph, A, B) = {x | ((x e. B -> ph) -> (x e. A /\ ph))}

Distinct variable groups:   ph,x   x,A   x,B

Proof of Theorem dfif2

Step	Hyp	Ref	Expression
1		df-if 3664	. 2 |- if(ph, A, B) = {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))}
2		df-or 359	. . . 4 |- (((x e. B /\ -. ph) \/ (x e. A /\ ph)) <-> (-. (x e. B /\ -. ph) -> (x e. A /\ ph)))
3		orcom 376	. . . 4 |- (((x e. A /\ ph) \/ (x e. B /\ -. ph)) <-> ((x e. B /\ -. ph) \/ (x e. A /\ ph)))
4		iman 413	. . . . 5 |- ((x e. B -> ph) <-> -. (x e. B /\ -. ph))
5	4	imbi1i 315	. . . 4 |- (((x e. B -> ph) -> (x e. A /\ ph)) <-> (-. (x e. B /\ -. ph) -> (x e. A /\ ph)))
6	2, 3, 5	3bitr4i 268	. . 3 |- (((x e. A /\ ph) \/ (x e. B /\ -. ph)) <-> ((x e. B -> ph) -> (x e. A /\ ph)))
7	6	abbii 2466	. 2 |- {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))} = {x | ((x e. B -> ph) -> (x e. A /\ ph))}
8	1, 7	eqtri 2373	1 |- if(ph, A, B) = {x | ((x e. B -> ph) -> (x e. A /\ ph))}

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  ifcif 3663

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-if 3664

This theorem is referenced by:  iftrue  3669  nfifd  3686