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Theorem ralcom4 2570
 Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
ralcom4 (x A yφyx A φ)
Distinct variable groups:   x,y   y,A
Allowed substitution hints:   φ(x,y)   A(x)

Proof of Theorem ralcom4
StepHypRef Expression
1 ralcom 2467 . 2 (x A y V φy V x A φ)
2 ralv 2565 . . 3 (y V φyφ)
32ralbii 2324 . 2 (x A y V φx A yφ)
4 ralv 2565 . 2 (y V x A φyx A φ)
51, 3, 43bitr3i 199 1 (x A yφyx A φ)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1240  ∀wral 2300  Vcvv 2551 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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553 This theorem is referenced by:  uniiunlem  3022  uni0b  3596  iunss  3689  trint  3860  reliun  4401  funimass4  5167  ralrnmpt2  5557
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