Theorem ralcom 2620

Description: Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
ralcom	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐴

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

Proof of Theorem ralcom

Step	Hyp	Ref	Expression
1		nfcv 2299	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2299	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	ralcomf 2618	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 104  ∀wral 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440

This theorem is referenced by:  ralcom4  2734  ssint  3823  issod  4279  reusv3  4420  cnvpom  5128  cnvsom  5129  fununi  5238  isocnv2  5762  dfsmo2  6234  ixpiinm  6669  rexfiuz  10889  tgss2  12490  cnmptcom  12709