Theorem nfra2 3154

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 39913. 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 2968	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 3149	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfral 3153	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  Ⅎwnf 1884  ∀wral 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121

This theorem is referenced by:  ralcom2  3313  invdisj  4858  reusv3  5104  dedekind  10518  dedekindle  10519  mreexexd  16660  gsummatr01lem4  20832  ordtconnlem1  30514  bnj1379  31446  tratrb  39579  islptre  40645  sprsymrelfo  42593