Theorem opelopabg

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 28-May-1995) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	opelopabg.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	opelopabg	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		opelopabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		opelopabg.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	opelopabga	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \chi \right)$$