Theorem r19.35OLD 3115

Description: Obsolete version of 19.35 1876 as of 22-Dec-2024. (Contributed by NM, 20-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem r19.35OLD

Step	Hyp	Ref	Expression
1		rexim 3093	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓) → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓))
2		pm2.27 42	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
3	2	ralimi 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓))
4	1, 3	syl11 33	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5		rexnal 3106	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
6		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
7	6	reximi 3090	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
8	5, 7	sylbir 235	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
9		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
10	9	reximi 3090	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
11	8, 10	ja 186	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
12	4, 11	impbii 209	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wral 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-ral 3068  df-rex 3077

This theorem is referenced by: (None)