Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by Mario Carneiro, 19-Dec-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | opelopaba.1 | ⊢ 𝐴 ∈ V | |
opelopaba.2 | ⊢ 𝐵 ∈ V | ||
opelopaba.3 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | ||
Assertion | opelopaba | ⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜓 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopaba.1 | ⊢ 𝐴 ∈ V | |
2 | opelopaba.2 | ⊢ 𝐵 ∈ V | |
3 | opelopaba.3 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | |
4 | 3 | opelopabga | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜓 ) ) |
5 | 1 2 4 | mp2an | ⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜓 ) |