Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-dfifc2 Structured version   Unicode version

Theorem bj-dfifc2 30728
Description: This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
Assertion
Ref Expression
bj-dfifc2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem bj-dfifc2
StepHypRef Expression
1 df-if 3886 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 ancom 448 . . . . 5  |-  ( (
ph  /\  x  e.  A )  <->  ( x  e.  A  /\  ph )
)
3 ancom 448 . . . . 5  |-  ( ( -.  ph  /\  x  e.  B )  <->  ( x  e.  B  /\  -.  ph ) )
42, 3orbi12i 519 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) )
54bicomi 202 . . 3  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) )  <->  ( ( ph  /\  x  e.  A
)  \/  ( -. 
ph  /\  x  e.  B ) ) )
65abbii 2536 . 2  |-  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
71, 6eqtri 2431 1  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   ifcif 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-if 3886
This theorem is referenced by:  bj-df-ifc  30729
  Copyright terms: Public domain W3C validator