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Theorem frege61c 43827
Description: Lemma for frege65c 43831. 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 43824 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege9 43715 . 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 1535  wcel 2103  [wsbc 3798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-frege1 43693  ax-frege2 43694  ax-frege8 43712  ax-frege58b 43804
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-sbc 3799
This theorem is referenced by:  frege65c  43831
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