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Theorem sbralie 2665
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  |-  ( A. x  e.  y  ph  <->  [ y  /  x ] A. y  e.  x  ps )
Distinct variable groups:    x, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem sbralie
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2663 . . 3  |-  ( A. y  e.  x  ps  <->  A. z  e.  x  [
z  /  y ] ps )
21sbbii 1738 . 2  |-  ( [ y  /  x ] A. y  e.  x  ps 
<->  [ y  /  x ] A. z  e.  x  [ z  /  y ] ps )
3 nfv 1508 . . 3  |-  F/ x A. z  e.  y  [ z  /  y ] ps
4 raleq 2624 . . 3  |-  ( x  =  y  ->  ( A. z  e.  x  [ z  /  y ] ps  <->  A. z  e.  y  [ z  /  y ] ps ) )
53, 4sbie 1764 . 2  |-  ( [ y  /  x ] A. z  e.  x  [ z  /  y ] ps  <->  A. z  e.  y  [ z  /  y ] ps )
6 cbvralsv 2663 . . 3  |-  ( A. z  e.  y  [
z  /  y ] ps  <->  A. x  e.  y  [ x  /  z ] [ z  /  y ] ps )
7 nfv 1508 . . . . . 6  |-  F/ z ps
87sbco2 1936 . . . . 5  |-  ( [ x  /  z ] [ z  /  y ] ps  <->  [ x  /  y ] ps )
9 nfv 1508 . . . . . 6  |-  F/ y
ph
10 sbralie.1 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110bicomd 140 . . . . . . 7  |-  ( x  =  y  ->  ( ps 
<-> 
ph ) )
1211equcoms 1684 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
139, 12sbie 1764 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
148, 13bitri 183 . . . 4  |-  ( [ x  /  z ] [ z  /  y ] ps  <->  ph )
1514ralbii 2439 . . 3  |-  ( A. x  e.  y  [
x  /  z ] [ z  /  y ] ps  <->  A. x  e.  y 
ph )
166, 15bitri 183 . 2  |-  ( A. z  e.  y  [
z  /  y ] ps  <->  A. x  e.  y 
ph )
172, 5, 163bitrri 206 1  |-  ( A. x  e.  y  ph  <->  [ y  /  x ] A. y  e.  x  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   [wsb 1735   A.wral 2414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419
This theorem is referenced by: (None)
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