Theorem r19.35

Description: Restricted quantifier version of 19.35 . (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Assertion	r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rexim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		pm2.27	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
3	2	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
4	1 3	syl11	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
5		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )
6		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
7	6	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
8	5 7	sylbir	⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
9		ax-1	⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) )
10	9	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
11	8 10	ja	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
12	4 11	impbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )