Theorem if0ab 15451

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 3613, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 15452 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 3454	. . . . . 6 |- -. x e. (/)
2	1	intnanr 931	. . . . 5 |- -. ( x e. (/) /\ -. ph )
3	2	biorfi 747	. . . 4 |- ( ( x e. A /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) )
4	3	bicomi 132	. . 3 |- ( ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) <-> ( x e. A /\ ph ) )
5	4	abbii 2312	. 2 |- { x | ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) } = { x | ( x e. A /\ ph ) }
6		df-if 3562	. 2 |- if ( ph , A , (/) ) = { x | ( ( x e. A /\ ph ) \/ ( x e. (/) /\ -. ph ) ) }
7		df-rab 2484	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8	5, 6, 7	3eqtr4i 2227	1 |- if ( ph , A , (/) ) = { x e. A | ph }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   {cab 2182   {crab 2479   (/)c0 3450   ifcif 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-dif 3159  df-nul 3451  df-if 3562

This theorem is referenced by: (None)