Theorem rabsnt

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006) (Proof shortened by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Hypotheses	rabsnt.1	$$⊢B\in \mathrm{V}$$
	rabsnt.2	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rabsnt	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		rabsnt.1	$$⊢B\in \mathrm{V}$$
2		rabsnt.2	$$⊢x=B\to \left(\phi \leftrightarrow \psi \right)$$
3	1	snid	$$⊢B\in \left\{B\right\}$$
4		id	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to \left\{x\in A|\phi \right\}=\left\{B\right\}$$
5	3 4	eleqtrrid	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to B\in \left\{x\in A|\phi \right\}$$
6	2	elrab	$$⊢B\in \left\{x\in A|\phi \right\}\leftrightarrow \left(B\in A\wedge \psi \right)$$
7	6	simprbi	$$⊢B\in \left\{x\in A|\phi \right\}\to \psi$$
8	5 7	syl	$$⊢\left\{x\in A|\phi \right\}=\left\{B\right\}\to \psi$$