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Theorem sbc2rexg 26882
Description: Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
sbc2rexg  |-  ( A  e.  V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
)
Distinct variable groups:    A, b    A, c    B, a    C, a   
a, b    a, c
Allowed substitution hints:    ph( a, b, c)    A( a)    B( b, c)    C( b, c)    V( a, b, c)

Proof of Theorem sbc2rexg
StepHypRef Expression
1 elex 2970 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcrexgOLD 3251 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  [. A  /  a ]. E. c  e.  C  ph )
)
3 sbcrexgOLD 3251 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  /  a ]. ph )
)
43rexbidv 2732 . . 3  |-  ( A  e.  _V  ->  ( E. b  e.  B  [. A  /  a ]. E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
)
52, 4bitrd 246 . 2  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
)
61, 5syl 16 1  |-  ( A  e.  V  ->  ( [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1727   E.wrex 2712   _Vcvv 2962   [.wsbc 3167
This theorem is referenced by:  sbc4rexg  26883  3rexfrabdioph  26895  7rexfrabdioph  26898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-v 2964  df-sbc 3168
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