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Theorem frege55c 36558
Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55c  |-  ( x  =  A  ->  A  =  x )

Proof of Theorem frege55c
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 3059 . . . 4  |-  x  e. 
_V
21frege54cor1c 36555 . . 3  |-  [. x  /  y ]. y  =  x
3 frege53c 36554 . . 3  |-  ( [. x  /  y ]. y  =  x  ->  ( x  =  A  ->  [. A  /  y ]. y  =  x ) )
42, 3ax-mp 5 . 2  |-  ( x  =  A  ->  [. A  /  y ]. y  =  x )
5 df-sbc 3279 . . . 4  |-  ( [. A  /  y ]. y  =  x  <->  A  e.  { y  |  y  =  x } )
6 clelab 2586 . . . 4  |-  ( A  e.  { y  |  y  =  x }  <->  E. y ( y  =  A  /\  y  =  x ) )
75, 6bitri 257 . . 3  |-  ( [. A  /  y ]. y  =  x  <->  E. y ( y  =  A  /\  y  =  x ) )
8 eqtr2 2481 . . . 4  |-  ( ( y  =  A  /\  y  =  x )  ->  A  =  x )
98exlimiv 1786 . . 3  |-  ( E. y ( y  =  A  /\  y  =  x )  ->  A  =  x )
107, 9sylbi 200 . 2  |-  ( [. A  /  y ]. y  =  x  ->  A  =  x )
114, 10syl 17 1  |-  ( x  =  A  ->  A  =  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447   _Vcvv 3056   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-frege8 36449  ax-frege52c 36528
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-sn 3980
This theorem is referenced by:  frege104  36607
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