Theorem nfra2w 3226

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41268. Version of nfra2 3227 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-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		nfcv 2976	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 3218	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfralw 3224	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  Ⅎwnf 1783  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142

This theorem is referenced by:  invdisj  5047  reusv3  5303  dedekind  10800  dedekindle  10801  mreexexd  16915  gsummatr01lem4  21263  ordtconnlem1  31191  bnj1379  32126  tratrb  40944  islptre  41974  sprsymrelfo  43733