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 ∈ V
	rabsnt.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rabsnt	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rabsnt.1	⊢ 𝐵 ∈ V
2		rabsnt.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
3	1	snid	⊢ 𝐵 ∈ { 𝐵 }
4		id	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } )
5	3 4	eleqtrrid	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )
6	2	elrab	⊢ ( 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ( 𝐵 ∈ 𝐴 ∧ 𝜓 ) )
7	6	simprbi	⊢ ( 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → 𝜓 )
8	5 7	syl	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝜓 )