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Theorem ifbi 3391
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 455 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
) )
2 id 19 . . . . . 6  |-  ( (
ph 
<->  ps )  ->  ( ph 
<->  ps ) )
32notbid 625 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
43anbi2d 452 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  B  /\  -.  ph )  <->  ( x  e.  B  /\  -.  ps ) ) )
51, 4orbi12d 740 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )  <-> 
( ( x  e.  A  /\  ps )  \/  ( x  e.  B  /\  -.  ps ) ) ) )
65abbidv 2200 . 2  |-  ( (
ph 
<->  ps )  ->  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) } )
7 df-if 3374 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
8 df-if 3374 . 2  |-  if ( ps ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) }
96, 7, 83eqtr4g 2140 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   {cab 2069   ifcif 3373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-if 3374
This theorem is referenced by:  ifbid  3392  ifbieq2i  3394  fodjuomni  6709  1tonninf  9735
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