Theorem dfif3 3533

Description: Alternate definition of the conditional operator df-if 3521. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)

Hypothesis

Ref	Expression
dfif3.1	|- C = { x | ph }

Assertion

Ref	Expression
dfif3	|- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem dfif3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif6 3522	. 2 |- if ( ph , A , B ) = ( { y e. A | ph } u. { y e. B | -. ph } )
2		dfif3.1	. . . . . 6 |- C = { x | ph }
3		biidd 171	. . . . . . 7 |- ( x = y -> ( ph <-> ph ) )
4	3	cbvabv 2291	. . . . . 6 |- { x | ph } = { y | ph }
5	2, 4	eqtri 2186	. . . . 5 |- C = { y | ph }
6	5	ineq2i 3320	. . . 4 |- ( A i^i C ) = ( A i^i { y | ph } )
7		dfrab3 3398	. . . 4 |- { y e. A | ph } = ( A i^i { y | ph } )
8	6, 7	eqtr4i 2189	. . 3 |- ( A i^i C ) = { y e. A | ph }
9		dfrab3 3398	. . . 4 |- { y e. B | -. ph } = ( B i^i { y | -. ph } )
10		notab 3392	. . . . . 6 |- { y | -. ph } = ( _V \ { y | ph } )
11	5	difeq2i 3237	. . . . . 6 |- ( _V \ C ) = ( _V \ { y | ph } )
12	10, 11	eqtr4i 2189	. . . . 5 |- { y | -. ph } = ( _V \ C )
13	12	ineq2i 3320	. . . 4 |- ( B i^i { y | -. ph } ) = ( B i^i ( _V \ C ) )
14	9, 13	eqtr2i 2187	. . 3 |- ( B i^i ( _V \ C ) ) = { y e. B | -. ph }
15	8, 14	uneq12i 3274	. 2 |- ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) = ( { y e. A | ph } u. { y e. B | -. ph } )
16	1, 15	eqtr4i 2189	1 |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Syntax hints:   -. wn 3   = wceq 1343   {cab 2151   {crab 2448   _Vcvv 2726   \ cdif 3113   u. cun 3114   i^i cin 3115   ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-if 3521

This theorem is referenced by: (None)