Theorem if0ab 3610

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. (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		dfif6 3609	. 2 |- if ( ph , A , (/) ) = ( { x e. A | ph } u. { x e. (/) | -. ph } )
2		rab0 3525	. . 3 |- { x e. (/) | -. ph } = (/)
3	2	uneq2i 3360	. 2 |- ( { x e. A | ph } u. { x e. (/) | -. ph } ) = ( { x e. A | ph } u. (/) )
4		un0 3530	. 2 |- ( { x e. A | ph } u. (/) ) = { x e. A | ph }
5	1, 3, 4	3eqtri 2256	1 |- if ( ph , A , (/) ) = { x e. A | ph }

Syntax hints:   -. wn 3   = wceq 1398   {crab 2515   u. cun 3199   (/)c0 3496   ifcif 3607

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-if 3608

This theorem is referenced by:  if0ss  3611