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Theorem nfrexd 2595
Description: Deduction version of nfrex 2598. (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 2556 . 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 1760 . . . 4  |-  ( ph  ->  F/ x  -.  ps )
62, 3, 5nfrald 2594 . . 3  |-  ( ph  ->  F/ x A. y  e.  A  -.  ps )
76nfnd 1760 . 2  |-  ( ph  ->  F/ x  -.  A. y  e.  A  -.  ps )
81, 7nfxfrd 1558 1  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   F/wnf 1531   F/_wnfc 2406   A.wral 2543   E.wrex 2544
This theorem is referenced by:  nfunid  3834
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  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
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