Theorem rspcedvd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv . (Contributed by AV, 27-Nov-2019)

	Ref	Expression
Hypotheses	rspcedvd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcedvd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	rspcedvd.3	⊢ ( 𝜑 → 𝜒 )
Assertion	rspcedvd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rspcedvd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		rspcedvd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		rspcedvd.3	⊢ ( 𝜑 → 𝜒 )
4	1 2	rspcedv	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )
5	3 4	mpd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )