Theorem brabga

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

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

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		brabga.2	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
3		df-br	$$⊢ARB\leftrightarrow ⟨A,B⟩\in R$$
4	2	eleq2i	$$⊢⟨A,B⟩\in R\leftrightarrow ⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}$$
5	3 4	bitri	$$⊢ARB\leftrightarrow ⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}$$
6	1	opelopabga	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \psi \right)$$
7	5 6	syl5bb	$$⊢\left(A\in V\wedge B\in W\right)\to \left(ARB\leftrightarrow \psi \right)$$