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Theorem nfrexdxy 2511
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2513 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 30-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
nfrexdxy  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem nfrexdxy
StepHypRef Expression
1 df-rex 2461 . 2  |-  ( E. y  e.  A  ps  <->  E. y ( y  e.  A  /\  ps )
)
2 nfraldxy.2 . . 3  |-  F/ y
ph
3 nfcv 2319 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfraldxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2335 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfraldxy.4 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfand 1568 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
92, 8nfexd 1761 . 2  |-  ( ph  ->  F/ x E. y
( y  e.  A  /\  ps ) )
101, 9nfxfrd 1475 1  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   F/wnf 1460   E.wex 1492    e. wcel 2148   F/_wnfc 2306   E.wrex 2456
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461
This theorem is referenced by:  nfrexdya  2513  nfrexxy  2516  nfunid  3814  strcollnft  14358
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