Theorem nfra2w 3299

Description: Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 45841. Version of nfra2 3362 with a disjoint variable condition not requiring ax-13 2402. (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 3199	. 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 1886	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2145  ∀wral 3077

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 1943  ax-10 2178  ax-11 2194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-ral 3078

This theorem is used by:  invdisj  5089  reusv3  5367  dedekind  11473  dedekindle  11474  mreexexd  17822  gsummatr01lem4  22973  vieta  34212  ordtconnlem1  34556  bnj1379  35460  tratrb  45518  islptre  46630  sprsymrelfo  48578