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Theorem sb8e 2065
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb8e  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )

Proof of Theorem sb8e
StepHypRef Expression
1 sb5rf.1 . . . . . 6  |-  F/ y
ph
21nfn 1793 . . . . 5  |-  F/ y  -.  ph
32sb8 2064 . . . 4  |-  ( A. x  -.  ph  <->  A. y [ y  /  x ]  -.  ph )
4 sbn 2034 . . . . 5  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
54albii 1557 . . . 4  |-  ( A. y [ y  /  x ]  -.  ph  <->  A. y  -.  [
y  /  x ] ph )
63, 5bitri 240 . . 3  |-  ( A. x  -.  ph  <->  A. y  -.  [
y  /  x ] ph )
76notbii 287 . 2  |-  ( -. 
A. x  -.  ph  <->  -. 
A. y  -.  [
y  /  x ] ph )
8 df-ex 1533 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
9 df-ex 1533 . 2  |-  ( E. y [ y  /  x ] ph  <->  -.  A. y  -.  [ y  /  x ] ph )
107, 8, 93bitr4i 268 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1531   E.wex 1532   F/wnf 1535   [wsb 1639
This theorem is referenced by:  exsbOLD  2103  sb8mo  2195  pm11.58  26737  bnj985  28496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640
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