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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
StepHypRef Expression
1 noel 3418 . . . . . 6  |-  -.  x  e.  (/)
21intnanr 925 . . . . 5  |-  -.  (
x  e.  (/)  /\  -.  ph )
32biorfi 741 . . . 4  |-  ( ( x  e.  A  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) )
43bicomi 131 . . 3  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  (/)  /\  -.  ph ) )  <->  ( x  e.  A  /\  ph )
)
54abbii 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 ) }
85, 6, 73eqtr4i 2201 1  |-  if (
ph ,  A ,  (/) )  =  { x  e.  A  |  ph }
Colors of variables: wff set class
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)
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