Description: The law of concretion. Theorem 9.5 of Quine p. 61. This version of opelopab uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013) (Proof shortened by Mario Carneiro, 18-Nov-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | opelopabaf.x | ⊢ Ⅎ 𝑥 𝜓 | |
opelopabaf.y | ⊢ Ⅎ 𝑦 𝜓 | ||
opelopabaf.1 | ⊢ 𝐴 ∈ V | ||
opelopabaf.2 | ⊢ 𝐵 ∈ V | ||
opelopabaf.3 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | ||
Assertion | opelopabaf | ⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜓 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabaf.x | ⊢ Ⅎ 𝑥 𝜓 | |
2 | opelopabaf.y | ⊢ Ⅎ 𝑦 𝜓 | |
3 | opelopabaf.1 | ⊢ 𝐴 ∈ V | |
4 | opelopabaf.2 | ⊢ 𝐵 ∈ V | |
5 | opelopabaf.3 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | |
6 | opelopabsb | ⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ) | |
7 | nfv | ⊢ Ⅎ 𝑥 𝐵 ∈ V | |
8 | 1 2 7 5 | sbc2iegf | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 ) ) |
9 | 3 4 8 | mp2an | ⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ 𝜓 ) |
10 | 6 9 | bitri | ⊢ ( 〈 𝐴 , 𝐵 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } ↔ 𝜓 ) |