Theorem nfra2xy 2529

Description: Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)

Assertion

Ref	Expression
nfra2xy	⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem nfra2xy

Step	Hyp	Ref	Expression
1		nfcv 2329	. 2 ⊢ Ⅎ𝑦𝐴
2		nfra1 2518	. 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfralxy 2525	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  Ⅎwnf 1470  ∀wral 2465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470

This theorem is referenced by:  invdisj  4009  reusv3  4472