Theorem nfra2 2669

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD in set.mm. 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 2490	. 2 ⊢ ℲyA
2		nfra1 2665	. 2 ⊢ Ⅎy∀y ∈ B φ
3	1, 2	nfral 2668	1 ⊢ Ⅎy∀x ∈ A ∀y ∈ B φ

Syntax hints:  Ⅎwnf 1544  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  ralcom2  2776  ncfinraise  4482