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Theorem frege68c 40563
 Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege68c ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))

Proof of Theorem frege68c
StepHypRef Expression
1 frege57aid 40504 . 2 ((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑))
2 frege59c.a . . 3 𝐴𝐵
32frege67c 40562 . 2 (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))
41, 3ax-mp 5 1 ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2114  [wsbc 3747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794  ax-frege1 40422  ax-frege2 40423  ax-frege8 40441  ax-frege52a 40489  ax-frege58b 40533 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-fal 1551  df-ex 1782  df-clab 2801  df-cleq 2815  df-clel 2894  df-sbc 3748 This theorem is referenced by:  frege70  40565  frege77  40572  frege116  40611
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