Theorem rexlimddv2

Description: Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	rexlimddv2.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
	rexlimddv2.2	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
Assertion	rexlimddv2	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		rexlimddv2.1	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
2		rexlimddv2.2	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
3	2	anasss	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
4	1 3	rexlimddv	⊢ ( 𝜑 → 𝜒 )