MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8e Unicode version

Theorem sb8e 2033
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 1765 . . . . 5  |-  F/ y  -.  ph
32sb8 2032 . . . 4  |-  ( A. x  -.  ph  <->  A. y [ y  /  x ]  -.  ph )
4 sbn 2002 . . . . 5  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
54albii 1553 . . . 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 1529 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
9 df-ex 1529 . 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 1527   E.wex 1528   F/wnf 1531   [wsb 1629
This theorem is referenced by:  exsbOLD  2070  sb8mo  2162  pm11.58  27589  bnj985  28985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
  Copyright terms: Public domain W3C validator