Theorem dfif2 2360

Description: An alternate definition of the conditional operator df-if 2359 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 2359	. 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 1573	. 2 |- {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))} = {x | ((x e. B -> ph) -> (x e. A /\ ph))}
8	1, 7	eqtr 1493	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 955   e. wcel 957  {cab 1462  ifcif 2358

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-if 2359

Copyright terms: Public domain