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Theorem sbcrexgOLD 35672
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3354 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrexgOLD  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
)
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem sbcrexgOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3281 . 2  |-  ( z  =  A  ->  ( [ z  /  x ] E. y  e.  B  ph  <->  [. A  /  x ]. E. y  e.  B  ph ) )
2 dfsbcq2 3281 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rexbidv 2912 . 2  |-  ( z  =  A  ->  ( E. y  e.  B  [ z  /  x ] ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
4 nfcv 2602 . . . 4  |-  F/_ x B
5 nfs1v 2276 . . . 4  |-  F/ x [ z  /  x ] ph
64, 5nfrex 2861 . . 3  |-  F/ x E. y  e.  B  [ z  /  x ] ph
7 sbequ12 2093 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87rexbidv 2912 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 2247 . 2  |-  ( [ z  /  x ] E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph )
101, 3, 9vtoclbg 3119 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454   [wsb 1807    e. wcel 1897   E.wrex 2749   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rex 2754  df-v 3058  df-sbc 3279
This theorem is referenced by:  2sbcrexOLD  35673  sbc2rexgOLD  35675
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