Metamath Proof Explorer


Theorem frege63c

Description: Analogue of frege63b . Proposition 63 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 𝐴𝐵
Assertion frege63c ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 𝐴𝐵
2 1 frege62c ( [ 𝐴 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) )
3 frege24 ( ( [ 𝐴 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )
4 2 3 ax-mp ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) ) )