Theorem r19.40

Description: Restricted quantifier version of Theorem 19.40 of Margaris p. 90. (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.40	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2	1	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐴 𝜑 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 )
4	3	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ∈ 𝐴 𝜓 )
5	2 4	jca	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) )