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Theorem acexmidlemph 5659
Description: Lemma for acexmid 5665. (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 668 . . . 4  |-  ( ph  ->  ( x  =  (/)  \/ 
ph ) )
21ralrimivw 2448 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
3 acexmidlem.a . . . . 5  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
43eqeq2i 2099 . . . 4  |-  ( {
(/) ,  { (/) } }  =  A  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
5 rabid2 2544 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <->  A. x  e.  { (/) ,  { (/) } }  (
x  =  (/)  \/  ph ) )
64, 5bitri 183 . . 3  |-  ( {
(/) ,  { (/) } }  =  A  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
72, 6sylibr 133 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  A )
8 olc 668 . . . 4  |-  ( ph  ->  ( x  =  { (/)
}  \/  ph )
)
98ralrimivw 2448 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
10 acexmidlem.b . . . . 5  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
1110eqeq2i 2099 . . . 4  |-  ( {
(/) ,  { (/) } }  =  B  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) } )
12 rabid2 2544 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
1311, 12bitri 183 . . 3  |-  ( {
(/) ,  { (/) } }  =  B  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
149, 13sylibr 133 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  B )
157, 14eqtr3d 2123 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 665    = wceq 1290   A.wral 2360   {crab 2364   (/)c0 3287   {csn 3450   {cpr 3451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-rab 2369
This theorem is referenced by:  acexmidlemab  5660
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