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Theorem sbc4rexg 26189
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 2872 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbc2rexg 26188 . . 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 26188 . . . 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 2662 . . 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 244 . 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 15 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 176    e. wcel 1710   E.wrex 2620   _Vcvv 2864   [.wsbc 3067
This theorem is referenced by:  6rexfrabdioph  26203  7rexfrabdioph  26204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-sbc 3068
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