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	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	opelopab2.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	opelopab2	$$⊢\left(A\in C\wedge B\in D\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\left(\left(x\in C\wedge y\in D\right)\wedge \phi \right)\right\}\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		opelopab2.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		opelopab2.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	sylan9bb	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \chi \right)$$
4	3	opelopab2a	$$⊢\left(A\in C\wedge B\in D\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\left(\left(x\in C\wedge y\in D\right)\wedge \phi \right)\right\}\leftrightarrow \chi \right)$$