Theorem 2alsraln0id 50545

Description: Nested general "all some" quantifiers with class membership as their antecedents, for the same class 𝐴: 𝜑 holds for every 𝑥 and every 𝑦 in 𝐴, and 𝐴 is not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)

Assertion

Ref	Expression
2alsraln0id	⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2alsraln0id

Step	Hyp	Ref	Expression
1		2alsraln0 50544	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅)))
2		pm4.24 573	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅))
3	2	bicomi 227	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ 𝐴 ≠ ∅)
4	3	anbi2i 634	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
5	1, 4	bitri 278	1 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2150   ≠ wne 2965  ∀wral 3086  ∅c0 4294  ∀∃wals 50513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-ne 2966  df-ral 3087  df-rex 3097  df-dif 3916  df-nul 4295  df-als 50515

This theorem is referenced by: (None)