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Theorem nfrald 2607
Description: Deduction version of nfral 2609. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-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
nfrald  |-  ( ph  ->  F/ x A. y  e.  A  ps )

Proof of Theorem nfrald
StepHypRef Expression
1 df-ral 2561 . 2  |-  ( A. y  e.  A  ps  <->  A. y ( y  e.  A  ->  ps )
)
2 nfrald.2 . . 3  |-  F/ y
ph
3 nfcvf 2454 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
43adantl 452 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
5 nfrald.3 . . . . . 6  |-  ( ph  -> 
F/_ x A )
65adantr 451 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
74, 6nfeld 2447 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
8 nfrald.4 . . . . 5  |-  ( ph  ->  F/ x ps )
98adantr 451 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
107, 9nfimd 1773 . . 3  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  ->  ps ) )
112, 10nfald2 1925 . 2  |-  ( ph  ->  F/ x A. y
( y  e.  A  ->  ps ) )
121, 11nfxfrd 1561 1  |-  ( ph  ->  F/ x A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   A.wral 2556
This theorem is referenced by:  nfrexd  2608  nfral  2609  riotasvdOLD  6364
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
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