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Theorem nfrexd 2608
Description: Deduction version of nfrex 2611. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfrald.2  |-  F/ y
ph
nfrald.3  |-  ( ph  -> 
F/_ x A )
nfrald.4  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfrexd  |-  ( ph  ->  F/ x E. y  e.  A  ps )

Proof of Theorem nfrexd
StepHypRef Expression
1 dfrex2 2569 . 2  |-  ( E. y  e.  A  ps  <->  -. 
A. y  e.  A  -.  ps )
2 nfrald.2 . . . 4  |-  F/ y
ph
3 nfrald.3 . . . 4  |-  ( ph  -> 
F/_ x A )
4 nfrald.4 . . . . 5  |-  ( ph  ->  F/ x ps )
54nfnd 1772 . . . 4  |-  ( ph  ->  F/ x  -.  ps )
62, 3, 5nfrald 2607 . . 3  |-  ( ph  ->  F/ x A. y  e.  A  -.  ps )
76nfnd 1772 . 2  |-  ( ph  ->  F/ x  -.  A. y  e.  A  -.  ps )
81, 7nfxfrd 1561 1  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1534   F/_wnfc 2419   A.wral 2556   E.wrex 2557
This theorem is referenced by:  nfunid  3850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562
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