Theorem dfif6 3640

Description: An alternate definition of the conditional operator df-if 3639 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfif6	|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

Distinct variable groups:   ph, x   x, A   x, B

Proof of Theorem dfif6

Step	Hyp	Ref	Expression
1		unab 3498	. 2 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
2		df-rab 2537	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2537	. . 3 |- { x e. B | -. ph } = { x | ( x e. B /\ -. ph ) }
4	2, 3	uneq12i 3381	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ -. ph ) } )
5		df-if 3639	. 2 |- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
6	1, 4, 5	3eqtr4ri 2270	1 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   u. cun 3218   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3639

This theorem is used by:  if0ab  3641  ifeq1  3643  ifeq2  3644  dfif3  3654  ifssun  3655