Metamath Proof Explorer


Theorem opelopab2

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007) (Revised by Mario Carneiro, 19-Dec-2013)

Ref Expression
Hypotheses opelopab2.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
opelopab2.2 ( 𝑦 = 𝐵 → ( 𝜓𝜒 ) )
Assertion opelopab2 ( ( 𝐴𝐶𝐵𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐶𝑦𝐷 ) ∧ 𝜑 ) } ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 opelopab2.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
2 opelopab2.2 ( 𝑦 = 𝐵 → ( 𝜓𝜒 ) )
3 1 2 sylan9bb ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( 𝜑𝜒 ) )
4 3 opelopab2a ( ( 𝐴𝐶𝐵𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥𝐶𝑦𝐷 ) ∧ 𝜑 ) } ↔ 𝜒 ) )