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Theorem nfraldxy 2442
Description: Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2444 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2  |-  F/ y
ph
nfraldxy.3  |-  ( ph  -> 
F/_ x A )
nfraldxy.4  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfraldxy  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem nfraldxy
StepHypRef Expression
1 df-ral 2396 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 nfraldxy.2 . . 3  |-  F/ y
ph
3 nfcv 2256 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfraldxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2272 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfraldxy.4 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfimd 1547 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  ->  ps ) )
92, 8nfald 1716 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
101, 9nfxfrd 1434 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312   F/wnf 1419    e. wcel 1463   F/_wnfc 2243   A.wral 2391
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by:  nfraldya  2444  nfralxy  2446
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