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Theorem 2sbcrex 26217
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 3027 . . . . 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 3008 . . 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 3027 . . 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 1621   E.wrex 2517   _Vcvv 2757   [.wsbc 2952
This theorem is referenced by:  2rexfrabdioph  26230  4rexfrabdioph  26232
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521  df-rex 2522  df-v 2759  df-sbc 2953
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