Theorem nfra2w 3270

Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 45042. Version of nfra2 3344 with a disjoint variable condition not requiring ax-13 2374. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by GG, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 3-Jan-2025.)

Assertion

Ref	Expression
nfra2w	⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

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

Proof of Theorem nfra2w

Step	Hyp	Ref	Expression
1		r2al 3170	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
2		nfa2 2181	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
3	1, 2	nfxfr 1854	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-10 2146  ax-11 2162

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-ral 3050

This theorem is referenced by:  invdisj  5082  reusv3  5348  dedekind  11294  dedekindle  11295  mreexexd  17569  gsummatr01lem4  22600  vieta  33685  ordtconnlem1  34030  bnj1379  34935  tratrb  44719  islptre  45807  sprsymrelfo  47685