Theorem nfra2 3154

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 42369. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfra2w 3151 when possible. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.)

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 2906	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 3142	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfral 3150	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  Ⅎwnf 1787  ∀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-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068

This theorem is referenced by:  ralcom2  3288