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Theorem frege61c 37735
 Description: Lemma for frege65c 37739. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege61c (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))

Proof of Theorem frege61c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege58c 37732 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege9 37623 . 2 ((∀𝑥𝜑[𝐴 / 𝑥]𝜑) → (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓)))
42, 3ax-mp 5 1 (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478   ∈ wcel 1987  [wsbc 3421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601  ax-frege1 37601  ax-frege2 37602  ax-frege8 37620  ax-frege58b 37712 This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1483  df-ex 1702  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191  df-sbc 3422 This theorem is referenced by:  frege65c  37739
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