Theorem braba

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

	Ref	Expression
Hypotheses	opelopaba.1	$$⊢A\in \mathrm{V}$$
	opelopaba.2	$$⊢B\in \mathrm{V}$$
	opelopaba.3	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
	braba.4	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
Assertion	braba	$$⊢ARB\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		opelopaba.1	$$⊢A\in \mathrm{V}$$
2		opelopaba.2	$$⊢B\in \mathrm{V}$$
3		opelopaba.3	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
4		braba.4	$$⊢R=\left\{⟨x,y⟩|\phi \right\}$$
5	3 4	brabga	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(ARB\leftrightarrow \psi \right)$$
6	1 2 5	mp2an	$$⊢ARB\leftrightarrow \psi$$