Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a particular context. Proposition 62 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | frege59c.a | ⊢ 𝐴 ∈ 𝐵 | |
| Assertion | frege62c | ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝐴 / 𝑥 ] 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege59c.a | ⊢ 𝐴 ∈ 𝐵 | |
| 2 | 1 | frege58c | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) ) |
| 3 | sbcim1 | ⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) | |
| 4 | 2 3 | syl | ⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) |
| 5 | ax-frege8 | ⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝐴 / 𝑥 ] 𝜓 ) ) ) | |
| 6 | 4 5 | ax-mp | ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝐴 / 𝑥 ] 𝜓 ) ) |