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Theorem frege61c 42270
Description: Lemma for frege65c 42274. 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 42267 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege9 42158 . 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 1540  wcel 2107  [wsbc 3744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-frege1 42136  ax-frege2 42137  ax-frege8 42155  ax-frege58b 42247
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3745
This theorem is referenced by:  frege65c  42274
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