Theorem reximdva0

Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012)

	Ref	Expression
Hypothesis	reximdva0.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )
Assertion	reximdva0	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		reximdva0.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )
2		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
3	1	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4	3	ancld	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
5	4	eximdv	⊢ ( 𝜑 → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
6	5	imp	⊢ ( ( 𝜑 ∧ ∃ 𝑥 𝑥 ∈ 𝐴 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7	2 6	sylan2b	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
8		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
9	7 8	sylibr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 𝜓 )