Theorem nfra2w 3273

Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 45167. Version of nfra2 3347 with a disjoint variable condition not requiring ax-13 2377. (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 3173	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
2		nfa2 2182	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
3	1, 2	nfxfr 1855	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-10 2147  ax-11 2163

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-ral 3053

This theorem is referenced by:  invdisj  5085  reusv3  5351  dedekind  11300  dedekindle  11301  mreexexd  17575  gsummatr01lem4  22606  vieta  33738  ordtconnlem1  34083  bnj1379  34988  tratrb  44844  islptre  45932  sprsymrelfo  47810