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Theorem sbralie 2947
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 2945 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21sbbii 1666 . . 3  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<->  [ x  /  y ] A. z  e.  y  [ z  /  x ] ph )
3 nfv 1630 . . . 4  |-  F/ y A. z  e.  x  [ z  /  x ] ph
4 raleq 2906 . . . 4  |-  ( y  =  x  ->  ( A. z  e.  y  [ z  /  x ] ph  <->  A. z  e.  x  [ z  /  x ] ph ) )
53, 4sbie 2152 . . 3  |-  ( [ x  /  y ] A. z  e.  y  [ z  /  x ] ph  <->  A. z  e.  x  [ z  /  x ] ph )
62, 5bitri 242 . 2  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<-> 
A. z  e.  x  [ z  /  x ] ph )
7 cbvralsv 2945 . . 3  |-  ( A. z  e.  x  [
z  /  x ] ph 
<-> 
A. y  e.  x  [ y  /  z ] [ z  /  x ] ph )
8 nfv 1630 . . . . . 6  |-  F/ z
ph
98sbco2 2163 . . . . 5  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
10 nfv 1630 . . . . . 6  |-  F/ x ps
11 sbralie.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1210, 11sbie 2152 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
139, 12bitri 242 . . . 4  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  ps )
1413ralbii 2731 . . 3  |-  ( A. y  e.  x  [
y  /  z ] [ z  /  x ] ph  <->  A. y  e.  x  ps )
157, 14bitri 242 . 2  |-  ( A. z  e.  x  [
z  /  x ] ph 
<-> 
A. y  e.  x  ps )
166, 15bitri 242 1  |-  ( [ x  /  y ] A. x  e.  y 
ph 
<-> 
A. y  e.  x  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   [wsb 1659   A.wral 2707
This theorem is referenced by:  tfinds2  4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712
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