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Theorem dvelimf 2010
Description: Version of dvelimv 1952 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
dvelimf.1  |-  F/ x ph
dvelimf.2  |-  F/ z ps
dvelimf.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimf  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )

Proof of Theorem dvelimf
StepHypRef Expression
1 dvelimf.2 . . . 4  |-  F/ z ps
2 dvelimf.3 . . . 4  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
31, 2equsal 1973 . . 3  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
43bicomi 193 . 2  |-  ( ps  <->  A. z ( z  =  y  ->  ph ) )
5 nfnae 1969 . . 3  |-  F/ z  -.  A. x  x  =  y
6 nfnae 1969 . . . . . 6  |-  F/ x  -.  A. x  x  =  y
7 nfnae 1969 . . . . . 6  |-  F/ x  -.  A. x  x  =  z
86, 7nfan 1834 . . . . 5  |-  F/ x
( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )
9 ax12o 1947 . . . . . 6  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
109impcom 419 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  ( z  =  y  ->  A. x  z  =  y )
)
118, 10nfd 1772 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  z  =  y )
12 dvelimf.1 . . . . 5  |-  F/ x ph
1312a1i 10 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x ph )
1411, 13nfimd 1815 . . 3  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x
( z  =  y  ->  ph ) )
155, 14nfald2 1985 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
164, 15nfxfrd 1576 1  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1545   F/wnf 1549
This theorem is referenced by:  dvelimnf  2030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550
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