Metamath Proof Explorer


Theorem frege61c

Description: Lemma for frege65c . Proposition 61 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 𝐴𝐵
Assertion frege61c ( ( [ 𝐴 / 𝑥 ] 𝜑𝜓 ) → ( ∀ 𝑥 𝜑𝜓 ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 𝐴𝐵
2 1 frege58c ( ∀ 𝑥 𝜑[ 𝐴 / 𝑥 ] 𝜑 )
3 frege9 ( ( ∀ 𝑥 𝜑[ 𝐴 / 𝑥 ] 𝜑 ) → ( ( [ 𝐴 / 𝑥 ] 𝜑𝜓 ) → ( ∀ 𝑥 𝜑𝜓 ) ) )
4 2 3 ax-mp ( ( [ 𝐴 / 𝑥 ] 𝜑𝜓 ) → ( ∀ 𝑥 𝜑𝜓 ) )