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Theorem ifbi 3758
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
StepHypRef Expression
1 dfbi3 865 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 iftrue 3747 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
3 iftrue 3747 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
43eqcomd 2443 . . . 4  |-  ( ps 
->  A  =  if ( ps ,  A ,  B ) )
52, 4sylan9eq 2490 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B )
)
6 iffalse 3748 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 iffalse 3748 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
87eqcomd 2443 . . . 4  |-  ( -. 
ps  ->  B  =  if ( ps ,  A ,  B ) )
96, 8sylan9eq 2490 . . 3  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B
) )
105, 9jaoi 370 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
111, 10sylbi 189 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653   ifcif 3741
This theorem is referenced by:  ifbid  3759  ifbieq2i  3760  dchrhash  21057  lgsdi  21118  rpvmasum2  21208  itg2gt0cn  26262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742
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