Theorem dfif2 2353

Description: An alternate definition of the conditional operator df-if 2352 with one fewer connectives (but probably less intuitive to understand).

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 2352	. 2 |- if(ph, A, B) = {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))}
2		df-or 224	. . . 4 |- (((x e. B /\ -. ph) \/ (x e. A /\ ph)) <-> (-. (x e. B /\ -. ph) -> (x e. A /\ ph)))
3		orcom 246	. . . 4 |- (((x e. A /\ ph) \/ (x e. B /\ -. ph)) <-> ((x e. B /\ -. ph) \/ (x e. A /\ ph)))
4		iman 237	. . . . 5 |- ((x e. B -> ph) <-> -. (x e. B /\ -. ph))
5	4	imbi1i 186	. . . 4 |- (((x e. B -> ph) -> (x e. A /\ ph)) <-> (-. (x e. B /\ -. ph) -> (x e. A /\ ph)))
6	2, 3, 5	3bitr4 183	. . 3 |- (((x e. A /\ ph) \/ (x e. B /\ -. ph)) <-> ((x e. B -> ph) -> (x e. A /\ ph)))
7	6	abbii 1567	. 2 |- {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))} = {x | ((x e. B -> ph) -> (x e. A /\ ph))}
8	1, 7	eqtr 1487	1 |- if(ph, A, B) = {x | ((x e. B -> ph) -> (x e. A /\ ph))}

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  ifcif 2351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

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