Theorem nfra2w 3301

Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 45596. Version of nfra2 3365 with a disjoint variable condition not requiring ax-13 2404. (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 3201	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
2		nfa2 2210	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
3	1, 2	nfxfr 1883	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1568  Ⅎwnf 1813   ∈ wcel 2143  ∀wral 3079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-10 2176  ax-11 2192

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-ral 3080

This theorem is used by:  invdisj  5095  reusv3  5376  dedekind  11377  dedekindle  11378  mreexexd  17708  gsummatr01lem4  22824  vieta  33979  ordtconnlem1  34323  bnj1379  35227  tratrb  45273  islptre  46363  sprsymrelfo  48274