ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifbi Unicode version

Theorem ifbi 3554
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 467 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
) )
2 notbi 666 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
32anbi2d 464 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  B  /\  -.  ph )  <->  ( x  e.  B  /\  -.  ps ) ) )
41, 3orbi12d 793 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )  <-> 
( ( x  e.  A  /\  ps )  \/  ( x  e.  B  /\  -.  ps ) ) ) )
54abbidv 2295 . 2  |-  ( (
ph 
<->  ps )  ->  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) } )
6 df-if 3535 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
7 df-if 3535 . 2  |-  if ( ps ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) }
85, 6, 73eqtr4g 2235 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   {cab 2163   ifcif 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-if 3535
This theorem is referenced by:  ifbid  3555  ifbieq2i  3557  fodjuomni  7144  fodjumkv  7155  nninfwlpoimlemg  7170  1tonninf  10435  lgsdi  14309  nninfsellemqall  14624  nninfomni  14628
  Copyright terms: Public domain W3C validator