Theorem if0ab 13840

Description: Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3574, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 13841 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

Assertion

Ref	Expression
if0ab	|- if ( ph , A , (/) ) = { x e. A | ph }

Distinct variable groups:   x, A   ph, x

Proof of Theorem if0ab

Step	Hyp	Ref	Expression
1		noel 3418	. . . . . 6 |- -. x e. (/)
2	1	intnanr 925	. . . . 5 |- -. ( x e. (/) /\ -. ph )
3	2	biorfi 741	. . . 4 |- ( ( x e. A /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) )
4	3	bicomi 131	. . 3 |- ( ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) <-> ( x e. A /\ ph ) )
5	4	abbii 2286	. 2 |- { x | ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) } = { x | ( x e. A /\ ph ) }
6		df-if 3527	. 2 |- if ( ph , A , (/) ) = { x | ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) }
7		df-rab 2457	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8	5, 6, 7	3eqtr4i 2201	1 |- if ( ph , A , (/) ) = { x e. A | ph }

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   (/)c0 3414   ifcif 3526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415  df-if 3527

This theorem is referenced by: (None)