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Theorem 2sbcrex 30646
Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
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 sbcrex 3421 . . 3  |-  ( [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. B  / 
b ]. ph )
21sbcbii 3396 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  [. A  / 
a ]. E. c  e.  C  [. B  / 
b ]. ph )
3 sbcrex 3421 . 2  |-  ( [. A  /  a ]. E. c  e.  C  [. B  /  b ]. ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
42, 3bitri 249 1  |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  / 
a ]. [. B  / 
b ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wrex 2818   [.wsbc 3336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120  df-sbc 3337
This theorem is referenced by:  2rexfrabdioph  30657  4rexfrabdioph  30659
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