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Theorem nfraldya 2510
Description: Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2508 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldya.2  |-  F/ y
ph
nfraldya.3  |-  ( ph  -> 
F/_ x A )
nfraldya.4  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfraldya  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x)

Proof of Theorem nfraldya
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ral 2458 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 sbim 1951 . . . . . 6  |-  ( [ z  /  y ] ( y  e.  A  ->  ps )  <->  ( [
z  /  y ] y  e.  A  ->  [ z  /  y ] ps ) )
3 clelsb1 2280 . . . . . . 7  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43imbi1i 238 . . . . . 6  |-  ( ( [ z  /  y ] y  e.  A  ->  [ z  /  y ] ps )  <->  ( z  e.  A  ->  [ z  /  y ] ps ) )
52, 4bitri 184 . . . . 5  |-  ( [ z  /  y ] ( y  e.  A  ->  ps )  <->  ( z  e.  A  ->  [ z  /  y ] ps ) )
65albii 1468 . . . 4  |-  ( A. z [ z  /  y ] ( y  e.  A  ->  ps )  <->  A. z ( z  e.  A  ->  [ z  /  y ] ps ) )
7 nfv 1526 . . . . 5  |-  F/ z ( y  e.  A  ->  ps )
87sb8 1854 . . . 4  |-  ( A. y ( y  e.  A  ->  ps )  <->  A. z [ z  / 
y ] ( y  e.  A  ->  ps ) )
9 df-ral 2458 . . . 4  |-  ( A. z  e.  A  [
z  /  y ] ps  <->  A. z ( z  e.  A  ->  [ z  /  y ] ps ) )
106, 8, 93bitr4i 212 . . 3  |-  ( A. y ( y  e.  A  ->  ps )  <->  A. z  e.  A  [
z  /  y ] ps )
11 nfv 1526 . . . 4  |-  F/ z
ph
12 nfraldya.3 . . . 4  |-  ( ph  -> 
F/_ x A )
13 nfraldya.2 . . . . 5  |-  F/ y
ph
14 nfraldya.4 . . . . 5  |-  ( ph  ->  F/ x ps )
1513, 14nfsbd 1975 . . . 4  |-  ( ph  ->  F/ x [ z  /  y ] ps )
1611, 12, 15nfraldxy 2508 . . 3  |-  ( ph  ->  F/ x A. z  e.  A  [ z  /  y ] ps )
1710, 16nfxfrd 1473 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
181, 17nfxfrd 1473 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1351   F/wnf 1458   [wsb 1760    e. wcel 2146   F/_wnfc 2304   A.wral 2453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458
This theorem is referenced by:  nfralya  2515
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