Theorem nfra2 3361

Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 45950. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker nfra2w 3298 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 2922	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 3286	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfral 3359	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

This proof depends on syntax axioms:  Ⅎwnf 1815  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077

This theorem is used by:  ralcom2  3362