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Theorem dvelimf 2137
Description: Version of dvelimv 2141 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
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 2095 . . 3  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
43bicomi 205 . 2  |-  ( ps  <->  A. z ( z  =  y  ->  ph ) )
5 nfnae 2119 . . 3  |-  F/ z  -.  A. x  x  =  y
6 nfeqf 2106 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  F/ x  z  =  y )
76ancoms 454 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  z  =  y )
8 dvelimf.1 . . . . 5  |-  F/ x ph
98a1i 11 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x ph )
107, 9nfimd 1977 . . 3  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x
( z  =  y  ->  ph ) )
115, 10nfald2 2134 . 2  |-  ( -. 
A. x  x  =  y  ->  F/ x A. z ( z  =  y  ->  ph ) )
124, 11nfxfrd 1691 1  |-  ( -. 
A. x  x  =  y  ->  F/ x ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  dvelimdf  2138  dvelimh  2139  dvelimnf  2142
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