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Theorem acexmidlemph 5861
Description: Lemma for acexmid 5867. (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 711 . . . 4  |-  ( ph  ->  ( x  =  (/)  \/ 
ph ) )
21ralrimivw 2551 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
3 acexmidlem.a . . . . 5  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
43eqeq2i 2188 . . . 4  |-  ( {
(/) ,  { (/) } }  =  A  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
5 rabid2 2653 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <->  A. x  e.  { (/) ,  { (/) } }  (
x  =  (/)  \/  ph ) )
64, 5bitri 184 . . 3  |-  ( {
(/) ,  { (/) } }  =  A  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  (/)  \/ 
ph ) )
72, 6sylibr 134 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  A )
8 olc 711 . . . 4  |-  ( ph  ->  ( x  =  { (/)
}  \/  ph )
)
98ralrimivw 2551 . . 3  |-  ( ph  ->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
10 acexmidlem.b . . . . 5  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
1110eqeq2i 2188 . . . 4  |-  ( {
(/) ,  { (/) } }  =  B  <->  { (/) ,  { (/) } }  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) } )
12 rabid2 2653 . . . 4  |-  ( {
(/) ,  { (/) } }  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
1311, 12bitri 184 . . 3  |-  ( {
(/) ,  { (/) } }  =  B  <->  A. x  e.  { (/)
,  { (/) } } 
( x  =  { (/)
}  \/  ph )
)
149, 13sylibr 134 . 2  |-  ( ph  ->  { (/) ,  { (/) } }  =  B )
157, 14eqtr3d 2212 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 708    = wceq 1353   A.wral 2455   {crab 2459   (/)c0 3422   {csn 3591   {cpr 3592
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-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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rab 2464
This theorem is referenced by:  acexmidlemab  5862
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