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 | ⊢ 𝐴 ∈ 𝐵 | |
Assertion | frege61c | ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c | ⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) |
3 | frege9 | ⊢ ( ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) → ( ( [ 𝐴 / 𝑥 ] 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) ) | |
4 | 2 3 | ax-mp | ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) |