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Theorem nfrald 2772
Description: Deduction version of nfral 2773. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfrald.1  |-  F/ y
ph
nfrald.2  |-  ( ph  -> 
F/_ x A )
nfrald.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfrald  |-  ( ph  ->  F/ x A. y  e.  A  ps )

Proof of Theorem nfrald
StepHypRef Expression
1 df-ral 2741 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 nfrald.1 . . 3  |-  F/ y
ph
3 nfcvf 2614 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
43adantl 468 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
5 nfrald.2 . . . . . 6  |-  ( ph  -> 
F/_ x A )
65adantr 467 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
74, 6nfeld 2599 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
8 nfrald.3 . . . . 5  |-  ( ph  ->  F/ x ps )
98adantr 467 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
107, 9nfimd 1999 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  ->  ps ) )
112, 10nfald2 2164 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
121, 11nfxfrd 1696 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371   A.wal 1441   F/wnf 1666    e. wcel 1886   F/_wnfc 2578   A.wral 2736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741
This theorem is referenced by:  nfral  2773  nfrexd  2848
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