Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of Quine p. 44. This is Frege's ninth axiom per Proposition 58 of Frege1879 p. 51. See also stdpc4 and rspsbc . (Contributed by NM, 16-Jan-2004)
Ref | Expression | ||
---|---|---|---|
Assertion | spsbc | ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 | ⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) | |
2 | sbsbc | ⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) | |
3 | 1 2 | sylib | ⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) |
4 | dfsbcq | ⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) ) | |
5 | 3 4 | syl5ib | ⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) ) |
6 | 5 | vtocleg | ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) ) |