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

Theorem acexmidlemph 5536
Description: Lemma for acexmid 5542. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
acexmidlem.b  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
acexmidlem.c  |-  C  =  { A ,  B }
Assertion
Ref Expression
acexmidlemph  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    x, B    x, C    ph, x

Proof of Theorem acexmidlemph
StepHypRef Expression
1 olc 665 . . . 4  |-  ( ph  ->  ( x  =  (/)  \/ 
ph ) )
21ralrimivw 2436 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
3 acexmidlem.a . . . . 5  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
43eqeq2i 2092 . . . 4  |-  ( {
(/) ,  { (/) } }  =  A  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
5 rabid2 2531 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <->  A. x  e.  { (/) ,  { (/) } }  (
x  =  (/)  \/  ph ) )
64, 5bitri 182 . . 3  |-  ( {
(/) ,  { (/) } }  =  A  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
72, 6sylibr 132 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  A )
8 olc 665 . . . 4  |-  ( ph  ->  ( x  =  { (/)
}  \/  ph )
)
98ralrimivw 2436 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
10 acexmidlem.b . . . . 5  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
1110eqeq2i 2092 . . . 4  |-  ( {
(/) ,  { (/) } }  =  B  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) } )
12 rabid2 2531 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
1311, 12bitri 182 . . 3  |-  ( {
(/) ,  { (/) } }  =  B  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
149, 13sylibr 132 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  B )
157, 14eqtr3d 2116 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285   A.wral 2349   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407
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-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 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-rab 2358
This theorem is referenced by:  acexmidlemab  5537
  Copyright terms: Public domain W3C validator