Theorem nfra2 3127

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 39856. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)

Assertion

Ref	Expression
nfra2	⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem nfra2

Step	Hyp	Ref	Expression
1		nfcv 2941	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 3122	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfral 3126	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  Ⅎwnf 1879  ∀wral 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094

This theorem is referenced by:  ralcom2  3285  invdisj  4829  reusv3  5075  dedekind  10490  dedekindle  10491  mreexexd  16623  gsummatr01lem4  20791  ordtconnlem1  30486  bnj1379  31418  tratrb  39522  islptre  40595  sprsymrelfo  42546