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Theorem frege53c 37125
Description: Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege53c ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵[𝐵 / 𝑥]𝜑))

Proof of Theorem frege53c
StepHypRef Expression
1 ax-frege52c 37099 . 2 (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
2 ax-frege8 37020 . 2 ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵[𝐵 / 𝑥]𝜑)))
31, 2ax-mp 5 1 ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵[𝐵 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  [wsbc 3306
This theorem was proved from axioms:  ax-mp 5  ax-frege8 37020  ax-frege52c 37099
This theorem is referenced by:  frege55lem2c  37128  frege55c  37129  frege56c  37130  frege92  37166
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