Theorem rspceeqv

Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022)

	Ref	Expression
Hypothesis	rspceeqv.1	⊢ ( 𝑥 = 𝐴 → 𝐶 = 𝐷 )
Assertion	rspceeqv	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝐸 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rspceeqv.1	⊢ ( 𝑥 = 𝐴 → 𝐶 = 𝐷 )
2	1	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( 𝐸 = 𝐶 ↔ 𝐸 = 𝐷 ) )
3	2	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝐸 = 𝐶 )