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Theorem rexcom4f 27199
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-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
rexcom4f  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem rexcom4f
StepHypRef Expression
1 ralcom4f.1 . . 3  |-  F/_ y A
2 nfcv 2629 . . 3  |-  F/_ x _V
31, 2rexcomf 3026 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. y  e.  _V  E. x  e.  A  ph )
4 rexv 3133 . . 3  |-  ( E. y  e.  _V  ph  <->  E. y ph )
54rexbii 2969 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. x  e.  A  E. y ph )
6 rexv 3133 . 2  |-  ( E. y  e.  _V  E. x  e.  A  ph  <->  E. y E. x  e.  A  ph )
73, 5, 63bitr3i 275 1  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1596   F/_wnfc 2615   E.wrex 2818   _Vcvv 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120
This theorem is referenced by: (None)
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