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Theorem ralcom4f 28096
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
Hypothesis
Ref Expression
ralcom4f.1  |-  F/_ y A
Assertion
Ref Expression
ralcom4f  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem ralcom4f
StepHypRef Expression
1 ralcom4f.1 . . 3  |-  F/_ y A
2 nfcv 2584 . . 3  |-  F/_ x _V
31, 2ralcomf 2987 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. y  e.  _V  A. x  e.  A  ph )
4 ralv 3095 . . 3  |-  ( A. y  e.  _V  ph  <->  A. y ph )
54ralbii 2856 . 2  |-  ( A. x  e.  A  A. y  e.  _V  ph  <->  A. x  e.  A  A. y ph )
6 ralv 3095 . 2  |-  ( A. y  e.  _V  A. x  e.  A  ph  <->  A. y A. x  e.  A  ph )
73, 5, 63bitr3i 278 1  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435   F/_wnfc 2570   A.wral 2775   _Vcvv 3081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-v 3083
This theorem is referenced by: (None)
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