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Theorem frege61c 40458
Description: Lemma for frege65c 40462. 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 40455 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege9 40346 . 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 1536  wcel 2115  [wsbc 3757
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796  ax-frege1 40324  ax-frege2 40325  ax-frege8 40343  ax-frege58b 40435
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clab 2803  df-cleq 2817  df-clel 2896  df-sbc 3758
This theorem is referenced by:  frege65c  40462
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