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Theorem ifbi 3460
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )

Proof of Theorem ifbi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anbi2 460 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
) )
2 notbi 638 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
32anbi2d 457 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  B  /\  -.  ph )  <->  ( x  e.  B  /\  -.  ps ) ) )
41, 3orbi12d 765 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )  <-> 
( ( x  e.  A  /\  ps )  \/  ( x  e.  B  /\  -.  ps ) ) ) )
54abbidv 2233 . 2  |-  ( (
ph 
<->  ps )  ->  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) } )
6 df-if 3443 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
7 df-if 3443 . 2  |-  if ( ps ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) }
85, 6, 73eqtr4g 2173 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   {cab 2101   ifcif 3442
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-if 3443
This theorem is referenced by:  ifbid  3461  ifbieq2i  3463  fodjuomni  6987  fodjumkv  7000  1tonninf  10153  nninfsellemqall  13013  nninfomni  13017
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