Theorem dfif2 4088

Description: An alternate definition of the conditional operator df-if 4087 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 4087	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-or 385	. . . 4 |- ( ( ( x e. B /\ -. ph ) \/ ( x e. A /\ ph ) ) <-> ( -. ( x e. B /\ -. ph ) -> ( x e. A /\ ph ) ) )
3		orcom 402	. . . 4 |- ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. B /\ -. ph ) \/ ( x e. A /\ ph ) ) )
4		iman 440	. . . . 5 |- ( ( x e. B -> ph ) <-> -. ( x e. B /\ -. ph ) )
5	4	imbi1i 339	. . . 4 |- ( ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) <-> ( -. ( x e. B /\ -. ph ) -> ( x e. A /\ ph ) ) )
6	2, 3, 5	3bitr4i 292	. . 3 |- ( ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) <-> ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) )
7	6	abbii 2739	. 2 |- { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } = { x | ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) }
8	1, 7	eqtri 2644	1 |- if ( ph , A , B ) = { x | ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) }

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   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-if 4087

This theorem is referenced by:  iftrue  4092  nfifd  4114