Theorem ralcom 2467

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 2175	. 2 ⊢ ℲyA
2		nfcv 2175	. 2 ⊢ ℲxB
3	1, 2	ralcomf 2465	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

Syntax hints:   ↔ wb 98  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305

This theorem is referenced by:  ralcom4  2570  ssint  3622  issod  4047  reusv3  4158  cnvpom  4803  cnvsom  4804  fununi  4910  isocnv2  5395  dfsmo2  5843