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Theorem rexcom4f 27943
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 2582 . . 3  |-  F/_ x _V
31, 2rexcomf 2986 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. y  e.  _V  E. x  e.  A  ph )
4 rexv 3093 . . 3  |-  ( E. y  e.  _V  ph  <->  E. y ph )
54rexbii 2925 . 2  |-  ( E. x  e.  A  E. y  e.  _V  ph  <->  E. x  e.  A  E. y ph )
6 rexv 3093 . 2  |-  ( E. y  e.  _V  E. x  e.  A  ph  <->  E. y E. x  e.  A  ph )
73, 5, 63bitr3i 278 1  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   E.wex 1659   F/_wnfc 2568   E.wrex 2774   _Vcvv 3078
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 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
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 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-v 3080
This theorem is referenced by: (None)
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