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Theorem if0ab 13422
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 3551,  |-  ( A  e.  V  ->  if ( ph ,  A ,  (/) )  e.  ~P A
), from which fmelpw1o 13423 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 3398 . . . . . 6  |-  -.  x  e.  (/)
21intnanr 916 . . . . 5  |-  -.  (
x  e.  (/)  /\  -.  ph )
32biorfi 736 . . . 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 2273 . 2  |-  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) }  =  { x  |  (
x  e.  A  /\  ph ) }
6 df-if 3506 . 2  |-  if (
ph ,  A ,  (/) )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) }
7 df-rab 2444 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
85, 6, 73eqtr4i 2188 1  |-  if (
ph ,  A ,  (/) )  =  { x  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 698    = wceq 1335    e. wcel 2128   {cab 2143   {crab 2439   (/)c0 3394   ifcif 3505
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-nul 3395  df-if 3506
This theorem is referenced by: (None)
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