Description: Principle related to sp . Axiom 58 of Frege1879 p. 51. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | frege58c.a | ⊢ 𝐴 ∈ 𝐵 | |
Assertion | frege58c | ⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege58c.a | ⊢ 𝐴 ∈ 𝐵 | |
2 | ax-frege58b | ⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) | |
3 | sbsbc | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) | |
4 | 2 3 | sylib | ⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) |
5 | dfsbcq | ⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) ) | |
6 | 4 5 | syl5ib | ⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) ) |
7 | 6 | vtocleg | ⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) ) |
8 | 1 7 | ax-mp | ⊢ ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) |