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Theorem nfexd2 2061
Description: Variation on nfexd 1873 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 1551 . 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 1809 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  -.  ps )
52, 4nfald2 2060 . . 3  |-  ( ph  ->  F/ x A. y  -.  ps )
65nfnd 1809 . 2  |-  ( ph  ->  F/ x  -.  A. y  -.  ps )
71, 6nfxfrd 1580 1  |-  ( ph  ->  F/ x E. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  nfeud2  2292  nfmod2  2293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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