Theorem reximia

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997) (Proof shortened by Wolf Lammen, 31-Oct-2024)

	Ref	Expression
Hypothesis	reximia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
Assertion	reximia	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		reximia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
2	1	imdistani	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	reximi2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )