Theorem bj-df-ifc 32565

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 2609. (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 32564	. 2 |- if ( ph , A , B ) = { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) }
2		df-ifp 1013	. . . 4 |- (if- ( ph , x e. A , x e. B ) <-> ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) )
3	2	bicomi 214	. . 3 |- ( ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) <-> if- ( ph , x e. A , x e. B ) )
4	3	abbii 2739	. 2 |- { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) } = { x | if- ( ph , x e. A , x e. B ) }
5	1, 4	eqtri 2644	1 |- if ( ph , A , B ) = { x | if- ( ph , x e. A , x e. B ) }

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384  if-wif 1012   = wceq 1483   e. wcel 1990   {cab 2608   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  bj-ififc  32566