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Theorem sbim 1870
Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sbim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)

Proof of Theorem sbim
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbimv 1816 . . . 4  |-  ( [ z  /  x ]
( ph  ->  ps )  <->  ( [ z  /  x ] ph  ->  [ z  /  x ] ps )
)
21sbbii 1690 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  z ] ( [ z  /  x ] ph  ->  [ z  /  x ] ps ) )
3 sbimv 1816 . . 3  |-  ( [ y  /  z ] ( [ z  /  x ] ph  ->  [ z  /  x ] ps ) 
<->  ( [ y  / 
z ] [ z  /  x ] ph  ->  [ y  /  z ] [ z  /  x ] ps ) )
42, 3bitri 182 . 2  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  ( [ y  /  z ] [
z  /  x ] ph  ->  [ y  / 
z ] [ z  /  x ] ps ) )
5 ax-17 1460 . . 3  |-  ( (
ph  ->  ps )  ->  A. z ( ph  ->  ps ) )
65sbco2v 1864 . 2  |-  ( [ y  /  z ] [ z  /  x ] ( ph  ->  ps )  <->  [ y  /  x ] ( ph  ->  ps ) )
7 ax-17 1460 . . . 4  |-  ( ph  ->  A. z ph )
87sbco2v 1864 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
9 ax-17 1460 . . . 4  |-  ( ps 
->  A. z ps )
109sbco2v 1864 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ps  <->  [ y  /  x ] ps )
118, 10imbi12i 237 . 2  |-  ( ( [ y  /  z ] [ z  /  x ] ph  ->  [ y  /  z ] [
z  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
124, 6, 113bitr3i 208 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbrim  1873  sblim  1874  sbbi  1876  moimv  2009  nfraldya  2405  sbcimg  2864  zfregfr  4344  tfi  4351  peano2  4364
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