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Theorem nfexd2 1913
Description: Variation on nfexd 1776 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfald2.1  |-  F/ y
ph
nfald2.2  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
Assertion
Ref Expression
nfexd2  |-  ( ph  ->  F/ x E. y ps )

Proof of Theorem nfexd2
StepHypRef Expression
1 df-ex 1529 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
2 nfald2.1 . . . 4  |-  F/ y
ph
3 nfald2.2 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
43nfnd 1760 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  -.  ps )
52, 4nfald2 1912 . . 3  |-  ( ph  ->  F/ x A. y  -.  ps )
65nfnd 1760 . 2  |-  ( ph  ->  F/ x  -.  A. y  -.  ps )
71, 6nfxfrd 1558 1  |-  ( ph  ->  F/ x E. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531
This theorem is referenced by:  nfeud2  2155  nfmod2  2156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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