Theorem 2alsraln0m 17325

Description: Nested general "all some" quantifiers with class membership as their antecedents: 𝜑 holds for every 𝑥 in 𝐴 and every 𝑦 in 𝐵, and both 𝐴 and 𝐵 are inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
2alsraln0m	⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑦)

Proof of Theorem 2alsraln0m

Step	Hyp	Ref	Expression
1		biid 171	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
2		alsraln0m 17321	. . . . 5 ⊢ (∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵))
3	1, 2	alsbii 17308	. . . 4 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ ∀∃𝑥(𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
4		alsraln0m 17321	. . . 4 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵)) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴))
5	3, 4	bitri 184	. . 3 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴))
6		r19.27mv 3624	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
7	6	pm5.32ri 459	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴))
8	5, 7	bitri 184	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴))
9		anass 405	. . 3 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑦 𝑦 ∈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴)))
10		ancom 266	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
11	10	anbi2i 461	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑦 𝑦 ∈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
12	9, 11	bitri 184	. 2 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 𝑦 ∈ 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))
13	8, 12	bitri 184	1 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵)))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∀∃wals 17293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-ral 2533  df-als 17295

This theorem is used by:  2alsraln0idm  17326