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Theorem 2sbcrex 26187
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 3142 . . . . 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 3123 . . 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 3142 . . 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 1710   E.wrex 2620   _Vcvv 2864   [.wsbc 3067
This theorem is referenced by:  2rexfrabdioph  26200  4rexfrabdioph  26202
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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