Theorem rspcedeq1vd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019)

	Ref	Expression
Hypotheses	rspcedeqvd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcedeqvd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐶 = 𝐷 )
Assertion	rspcedeq1vd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		rspcedeqvd.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐶 = 𝐷 )
3	2	eqeq1d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝐶 = 𝐷 ↔ 𝐷 = 𝐷 ) )
4		eqidd	⊢ ( 𝜑 → 𝐷 = 𝐷 )
5	1 3 4	rspcedvd	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐶 = 𝐷 )