Theorem dfif3 3518

Description: Alternate definition of the conditional operator df-if 3506. 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 3507	. 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 2282	. . . . . 6 |- { x | ph } = { y | ph }
5	2, 4	eqtri 2178	. . . . 5 |- C = { y | ph }
6	5	ineq2i 3305	. . . 4 |- ( A i^i C ) = ( A i^i { y | ph } )
7		dfrab3 3383	. . . 4 |- { y e. A | ph } = ( A i^i { y | ph } )
8	6, 7	eqtr4i 2181	. . 3 |- ( A i^i C ) = { y e. A | ph }
9		dfrab3 3383	. . . 4 |- { y e. B | -. ph } = ( B i^i { y | -. ph } )
10		notab 3377	. . . . . 6 |- { y | -. ph } = ( _V \ { y | ph } )
11	5	difeq2i 3222	. . . . . 6 |- ( _V \ C ) = ( _V \ { y | ph } )
12	10, 11	eqtr4i 2181	. . . . 5 |- { y | -. ph } = ( _V \ C )
13	12	ineq2i 3305	. . . 4 |- ( B i^i { y | -. ph } ) = ( B i^i ( _V \ C ) )
14	9, 13	eqtr2i 2179	. . 3 |- ( B i^i ( _V \ C ) ) = { y e. B | -. ph }
15	8, 14	uneq12i 3259	. 2 |- ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) = ( { y e. A | ph } u. { y e. B | -. ph } )
16	1, 15	eqtr4i 2181	1 |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Syntax hints:   -. wn 3   = wceq 1335   {cab 2143   {crab 2439   _Vcvv 2712   \ cdif 3099   u. cun 3100   i^i cin 3101   ifcif 3505

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-if 3506

This theorem is referenced by: (None)