Theorem 2alsraln0id 51192

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 51191	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅)))
2		pm4.24 574	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅))
3	2	bicomi 227	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ 𝐴 ≠ ∅)
4	3	anbi2i 635	. 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 ≠ ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∅c0 4281  ∀∃wals 51160

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-ne 2960  df-ral 3081  df-rex 3091  df-dif 4050  df-nul 4282  df-als 51162

This theorem is used by: (None)