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Theorem sbc2rexg 26865
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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcrexg 3066 . . 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 sbcrexg 3066 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  /  a ]. ph )
)
43rexbidv 2564 . . 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 244 . 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 15 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 176    e. wcel 1684   E.wrex 2544   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  sbc4rexg  26866  3rexfrabdioph  26878  7rexfrabdioph  26881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992
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