Description: The law of concretion. Special case of Theorem 9.5 of Quine p. 61. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker opabidw when possible. (Contributed by NM, 14-Apr-1995) (Proof shortened by Andrew Salmon, 25-Jul-2011) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | opabid | ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opex | ⊢ ⟨ 𝑥 , 𝑦 ⟩ ∈ V | |
| 2 | copsexg | ⊢ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) | |
| 3 | 2 | bicomd | ⊢ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜑 ) ) |
| 4 | df-opab | ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } | |
| 5 | 1 3 4 | elab2 | ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜑 ) |