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Theorem frege56c 36559
Description: Lemma for frege57c 36560. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege56c.b  |-  B  e.  C
Assertion
Ref Expression
frege56c  |-  ( ( A  =  B  -> 
( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )  ->  ( B  =  A  ->  ( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    C( x)

Proof of Theorem frege56c
StepHypRef Expression
1 frege56c.b . . . . 5  |-  B  e.  C
21frege54cor1c 36555 . . . 4  |-  [. B  /  x ]. x  =  B
3 frege53c 36554 . . . 4  |-  ( [. B  /  x ]. x  =  B  ->  ( B  =  A  ->  [. A  /  x ]. x  =  B ) )
42, 3ax-mp 5 . . 3  |-  ( B  =  A  ->  [. A  /  x ]. x  =  B )
5 frege55lem1c 36556 . . 3  |-  ( ( B  =  A  ->  [. A  /  x ]. x  =  B
)  ->  ( B  =  A  ->  A  =  B ) )
64, 5ax-mp 5 . 2  |-  ( B  =  A  ->  A  =  B )
7 frege9 36452 . 2  |-  ( ( B  =  A  ->  A  =  B )  ->  ( ( A  =  B  ->  ( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )  ->  ( B  =  A  ->  (
[. A  /  x ]. ph  ->  [. B  /  x ]. ph ) ) ) )
86, 7ax-mp 5 1  |-  ( ( A  =  B  -> 
( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )  ->  ( B  =  A  ->  ( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   [.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-frege1 36430  ax-frege2 36431  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:  frege57c  36560
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