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Theorem 2sbcrex 26967
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 3079 . . . . 5  |-  ( B  e.  _V  ->  ( [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. B  /  b ]. ph )
)
42, 3ax-mp 8 . . . 4  |-  ( [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. B  / 
b ]. ph )
54sbcbiiOLD 3060 . . 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 8 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  [. A  / 
a ]. E. c  e.  C  [. B  / 
b ]. ph )
7 sbcrexg 3079 . . 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 8 . 2  |-  ( [. A  /  a ]. E. c  e.  C  [. B  /  b ]. ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
96, 8bitri 240 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 176    e. wcel 1696   E.wrex 2557   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  2rexfrabdioph  26980  4rexfrabdioph  26982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005
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