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Theorem sbralie 2603
Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
Hypothesis
Ref Expression
sbralie.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbralie  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<-> 
A. y  e.  x  ps )
Distinct variable groups:    x, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem sbralie
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2601 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21sbbii 1695 . . 3  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<->  [ x  /  y ] A. z  e.  y  [ z  /  x ] ph )
3 nfv 1466 . . . 4  |-  F/ y A. z  e.  x  [ z  /  x ] ph
4 raleq 2562 . . . 4  |-  ( y  =  x  ->  ( A. z  e.  y  [ z  /  x ] ph  <->  A. z  e.  x  [ z  /  x ] ph ) )
53, 4sbie 1721 . . 3  |-  ( [ x  /  y ] A. z  e.  y  [ z  /  x ] ph  <->  A. z  e.  x  [ z  /  x ] ph )
62, 5bitri 182 . 2  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<-> 
A. z  e.  x  [ z  /  x ] ph )
7 cbvralsv 2601 . . 3  |-  ( A. z  e.  x  [
z  /  x ] ph 
<-> 
A. y  e.  x  [ y  /  z ] [ z  /  x ] ph )
8 nfv 1466 . . . . . 6  |-  F/ z
ph
98sbco2 1887 . . . . 5  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
10 nfv 1466 . . . . . 6  |-  F/ x ps
11 sbralie.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1210, 11sbie 1721 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
139, 12bitri 182 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  ps )
1413ralbii 2384 . . 3  |-  ( A. y  e.  x  [
y  /  z ] [ z  /  x ] ph  <->  A. y  e.  x  ps )
157, 14bitri 182 . 2  |-  ( A. z  e.  x  [
z  /  x ] ph 
<-> 
A. y  e.  x  ps )
166, 15bitri 182 1  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<-> 
A. y  e.  x  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   [wsb 1692   A.wral 2359
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364
This theorem is referenced by: (None)
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