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Theorem frege53c 36554
Description: Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege53c  |-  ( [. A  /  x ]. ph  ->  ( A  =  B  ->  [. B  /  x ]. ph ) )

Proof of Theorem frege53c
StepHypRef Expression
1 ax-frege52c 36528 . 2  |-  ( A  =  B  ->  ( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )
2 ax-frege8 36449 . 2  |-  ( ( A  =  B  -> 
( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )  ->  ( [. A  /  x ]. ph  ->  ( A  =  B  ->  [. B  /  x ]. ph ) ) )
31, 2ax-mp 5 1  |-  ( [. A  /  x ]. ph  ->  ( A  =  B  ->  [. B  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-frege8 36449  ax-frege52c 36528
This theorem is referenced by:  frege55lem2c  36557  frege55c  36558  frege56c  36559  frege92  36595
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