Theorem bj-df-ifc 31159

Description: The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2438. (Contributed by BJ, 20-Sep-2019.)

Assertion

Ref	Expression
bj-df-ifc	|- if ( ph , A , B ) = { x | if- ( ph , x e. A , x e. B ) }

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

Proof of Theorem bj-df-ifc

Step	Hyp	Ref	Expression
1		bj-dfifc2 31158	. 2 |- if ( ph , A , B ) = { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) }
2		df-ifp 1426	. . . 4 |- (if- ( ph , x e. A , x e. B ) <-> ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) )
3	2	bicomi 206	. . 3 |- ( ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) <-> if- ( ph , x e. A , x e. B ) )
4	3	abbii 2567	. 2 |- { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) } = { x | if- ( ph , x e. A , x e. B ) }
5	1, 4	eqtri 2473	1 |- if ( ph , A , B ) = { x | if- ( ph , x e. A , x e. B ) }

Syntax hints:   -. wn 3   \/ wo 370   /\ wa 371  if-wif 1425   = wceq 1444   e. wcel 1887   {cab 2437   ifcif 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ifp 1426  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-if 3882

This theorem is referenced by:  bj-ififc  31160