Metamath Proof Explorer

Theorem frege66b

Description: Swap antecedents of frege65b . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Assertion frege66b ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝑦 / 𝑥 ] 𝜒 → [ 𝑦 / 𝑥 ] 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 frege65b ⊢ ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜒 → [ 𝑦 / 𝑥 ] 𝜓 ) ) )
2 ax-frege8 ⊢ ( ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜒 → [ 𝑦 / 𝑥 ] 𝜓 ) ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝑦 / 𝑥 ] 𝜒 → [ 𝑦 / 𝑥 ] 𝜓 ) ) ) )
3 1 2 ax-mp ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝑦 / 𝑥 ] 𝜒 → [ 𝑦 / 𝑥 ] 𝜓 ) ) )