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Theorem sbc4rexg 26735
Description: Exchange a substitution with 4 existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
sbc4rexg  |-  ( A  e.  V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph )
)
Distinct variable groups:    A, b    A, c    B, a    C, a   
a, b    a, c    A, d    A, e    D, a    E, a    a, d    e,
a
Allowed substitution hints:    ph( e, a, b, c, d)    A( a)    B( e, b, c, d)    C( e, b, c, d)    D( e, b, c, d)    E( e, b, c, d)    V( e, a, b, c, d)

Proof of Theorem sbc4rexg
StepHypRef Expression
1 elex 2924 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbc2rexg 26734 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. E. d  e.  D  E. e  e.  E  ph )
)
3 sbc2rexg 26734 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. d  e.  D  E. e  e.  E  ph  <->  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph )
)
432rexbidv 2709 . . 3  |-  ( A  e.  _V  ->  ( E. b  e.  B  E. c  e.  C  [. A  /  a ]. E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph )
)
52, 4bitrd 245 . 2  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph )
)
61, 5syl 16 1  |-  ( A  e.  V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   E.wrex 2667   _Vcvv 2916   [.wsbc 3121
This theorem is referenced by:  6rexfrabdioph  26749  7rexfrabdioph  26750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-sbc 3122
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