Theorem rspcdv

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007) (Revised by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rspcdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	biimpd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) )
4	1 3	rspcimdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )