Theorem raaanv 3567

Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Assertion

Ref	Expression
raaanv	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem raaanv

Step	Hyp	Ref	Expression
1		nfv 1551	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1551	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	raaan 3566	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wral 2484

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  reusv3i  4506  f1mpt  5840