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Theorem 2sbcrex 26263
Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Hypotheses
Ref Expression
2sbcrex.1  |-  A  e. 
_V
2sbcrex.2  |-  B  e. 
_V
Assertion
Ref Expression
2sbcrex  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
Distinct variable groups:    A, c    B, c    C, b    a, c   
b, c    C, a
Allowed substitution hints:    ph( a, b, c)    A( a, b)    B( a, b)    C( c)

Proof of Theorem 2sbcrex
StepHypRef Expression
1 2sbcrex.1 . . 3  |-  A  e. 
_V
2 2sbcrex.2 . . . . 5  |-  B  e. 
_V
3 sbcrexg 3067 . . . . 5  |-  ( B  e.  _V  ->  ( [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. B  /  b ]. ph )
)
42, 3ax-mp 10 . . . 4  |-  ( [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. B  / 
b ]. ph )
54sbcbiiOLD 3048 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  [. A  /  a ]. E. c  e.  C  [. B  /  b ]. ph ) )
61, 5ax-mp 10 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  [. A  / 
a ]. E. c  e.  C  [. B  / 
b ]. ph )
7 sbcrexg 3067 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  a ]. E. c  e.  C  [. B  /  b ]. ph  <->  E. c  e.  C  [. A  /  a ]. [. B  /  b ]. ph )
)
81, 7ax-mp 10 . 2  |-  ( [. A  /  a ]. E. c  e.  C  [. B  /  b ]. ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
96, 8bitri 242 1  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1685   E.wrex 2545   _Vcvv 2789   [.wsbc 2992
This theorem is referenced by:  2rexfrabdioph  26276  4rexfrabdioph  26278
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-v 2791  df-sbc 2993
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