Theorem bj-df-ifc 33123

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 2448. (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 33122	. 2 |- if ( ph , A , B ) = { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) }
2		bj-dfif5 33106	. . . 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 2596	. 2 |- { x | ( ( ph /\ x e. A ) \/ ( -. ph /\ x e. B ) ) } = { x | if- ( ph , x e. A , x e. B ) }
5	1, 4	eqtri 2491	1 |- if ( ph , A , B ) = { x | if- ( ph , x e. A , x e. B ) }

Syntax hints:   -. wn 3   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2447   ifcif 3934  if-wif 33101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-if 3935  df-bj-if 33102

This theorem is referenced by:  bj-ififc  33124