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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 φy 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
StepHypRef Expression
1 nfcv 2175 . 2 yA
2 nfcv 2175 . 2 xB
31, 2ralcomf 2465 1 (x A y B φy B x A φ)
Colors of variables: wff set class
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
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