Theorem brabg

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	opelopabg.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
	brabg.5	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
Assertion	brabg	$$⊢\left(A\in C\wedge B\in D\right)\to \left(ARB\leftrightarrow \chi \right)$$

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		brabg.5	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
4	1 2	sylan9bb	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \chi \right)$$
5	4 3	brabga	$$⊢\left(A\in C\wedge B\in D\right)\to \left(ARB\leftrightarrow \chi \right)$$