Theorem nfra2w 3151

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 42369. Version of nfra2 3154 with a disjoint variable condition not requiring ax-13 2372. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)

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 3124	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
2		alcom 2158	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
3	1, 2	bitri 274	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
4		nfa1 2150	. 2 ⊢ Ⅎ𝑦∀𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
5	3, 4	nfxfr 1856	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  Ⅎwnf 1787   ∈ wcel 2108  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-10 2139  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-ral 3068

This theorem is referenced by:  invdisj  5054  reusv3  5323  dedekind  11068  dedekindle  11069  mreexexd  17274  gsummatr01lem4  21715  ordtconnlem1  31776  bnj1379  32710  tratrb  42045  islptre  43050  sprsymrelfo  44837