Theorem rmobidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017)

	Ref	Expression
Hypothesis	rmobidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rmobidva	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rmobidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜑
3	2 1	rmobida	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐴 𝜒 ) )