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Theorem nfraldw 2496
Description: Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2499 for a version with  y and  A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfraldw.1  |-  F/ y
ph
nfraldw.2  |-  ( ph  -> 
F/_ x A )
nfraldw.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfraldw  |-  ( 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 nfraldw
StepHypRef Expression
1 df-ral 2447 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 nfraldw.1 . . 3  |-  F/ y
ph
3 nfcvd 2307 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfraldw.2 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeld 2322 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
6 nfraldw.3 . . . 4  |-  ( ph  ->  F/ x ps )
75, 6nfimd 1572 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  ->  ps ) )
82, 7nfald 1747 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
91, 8nfxfrd 1462 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1340   F/wnf 1447    e. wcel 2135   F/_wnfc 2293   A.wral 2442
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447
This theorem is referenced by:  nfralw  2501
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