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Theorem frege67c 39063
 Description: Lemma for frege68c 39064. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege67c (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))

Proof of Theorem frege67c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege58c 39054 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege7 38941 . 2 ((∀𝑥𝜑[𝐴 / 𝑥]𝜑) → (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))))
42, 3ax-mp 5 1 (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654   ∈ wcel 2164  [wsbc 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-12 2220  ax-ext 2803  ax-frege1 38923  ax-frege2 38924  ax-frege58b 39034 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663 This theorem is referenced by:  frege68c  39064
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