Theorem r19.35

Description: Restricted quantifier version of 19.35 . (Contributed by NM, 20-Sep-2003) (Proof shortened by Wolf Lammen, 22-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		pm5.5	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) ↔ 𝜓 ) )
2	1	ralrexbid	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3	2	biimpcd	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
4		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )
5		pm2.21	⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) )
6	5	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
7	4 6	sylbir	⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
8		ax-1	⊢ ( 𝜓 → ( 𝜑 → 𝜓 ) )
9	8	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
10	7 9	ja	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) )
11	3 10	impbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )