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Theorem nfraldya 2375
Description: Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2373 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 2328 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 sbim 1843 . . . . . 6  |-  ( [ z  /  y ] ( y  e.  A  ->  ps )  <->  ( [
z  /  y ] y  e.  A  ->  [ z  /  y ] ps ) )
3 clelsb3 2158 . . . . . . 7  |-  ( [ z  /  y ] y  e.  A  <->  z  e.  A )
43imbi1i 231 . . . . . 6  |-  ( ( [ z  /  y ] y  e.  A  ->  [ z  /  y ] ps )  <->  ( z  e.  A  ->  [ z  /  y ] ps ) )
52, 4bitri 177 . . . . 5  |-  ( [ z  /  y ] ( y  e.  A  ->  ps )  <->  ( z  e.  A  ->  [ z  /  y ] ps ) )
65albii 1375 . . . 4  |-  ( A. z [ z  /  y ] ( y  e.  A  ->  ps )  <->  A. z ( z  e.  A  ->  [ z  /  y ] ps ) )
7 nfv 1437 . . . . 5  |-  F/ z ( y  e.  A  ->  ps )
87sb8 1752 . . . 4  |-  ( A. y ( y  e.  A  ->  ps )  <->  A. z [ z  / 
y ] ( y  e.  A  ->  ps ) )
9 df-ral 2328 . . . 4  |-  ( A. z  e.  A  [
z  /  y ] ps  <->  A. z ( z  e.  A  ->  [ z  /  y ] ps ) )
106, 8, 93bitr4i 205 . . 3  |-  ( A. y ( y  e.  A  ->  ps )  <->  A. z  e.  A  [
z  /  y ] ps )
11 nfv 1437 . . . 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 1867 . . . 4  |-  ( ph  ->  F/ x [ z  /  y ] ps )
1611, 12, 15nfraldxy 2373 . . 3  |-  ( ph  ->  F/ x A. z  e.  A  [ z  /  y ] ps )
1710, 16nfxfrd 1380 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
181, 17nfxfrd 1380 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   F/wnf 1365    e. wcel 1409   [wsb 1661   F/_wnfc 2181   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by:  nfralya  2379
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