Theorem bj-df-ifc 30716

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 2388. (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 30715	. 2 |- if ( ph , A , B ) = { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) }
2		df-ifp 1387	. . . 4 |- (if- ( ph , x e. A , x e. B ) <-> ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) )
3	2	bicomi 202	. . 3 |- ( ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) <-> if- ( ph , x e. A , x e. B ) )
4	3	abbii 2536	. 2 |- { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) } = { x | if- ( ph , x e. A , x e. B ) }
5	1, 4	eqtri 2431	1 |- if ( ph , A , B ) = { x | if- ( ph , x e. A , x e. B ) }

Syntax hints:   -. wn 3   \/ wo 366   /\ wa 367  if-wif 1386   = wceq 1405   e. wcel 1842   {cab 2387   ifcif 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ifp 1387  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-if 3885

This theorem is referenced by:  bj-ififc  30717