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Theorem cbvrexf 3083
Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1  |-  F/_ x A
cbvralf.2  |-  F/_ y A
cbvralf.3  |-  F/ y
ph
cbvralf.4  |-  F/ x ps
cbvralf.5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrexf  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )

Proof of Theorem cbvrexf
StepHypRef Expression
1 cbvralf.1 . . . 4  |-  F/_ x A
2 cbvralf.2 . . . 4  |-  F/_ y A
3 cbvralf.3 . . . . 5  |-  F/ y
ph
43nfn 1849 . . . 4  |-  F/ y  -.  ph
5 cbvralf.4 . . . . 5  |-  F/ x ps
65nfn 1849 . . . 4  |-  F/ x  -.  ps
7 cbvralf.5 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
87notbid 294 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
91, 2, 4, 6, 8cbvralf 3082 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. y  e.  A  -.  ps )
109notbii 296 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. y  e.  A  -.  ps )
11 dfrex2 2915 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
12 dfrex2 2915 . 2  |-  ( E. y  e.  A  ps  <->  -. 
A. y  e.  A  -.  ps )
1310, 11, 123bitr4i 277 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   F/wnf 1599   F/_wnfc 2615   A.wral 2814   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820
This theorem is referenced by:  cbvrex  3085  reusv2lem4  4651  reusv2  4653  nnwof  11148  dfimafnf  27174  indexa  29855  evth2f  30996  fvelrnbf  30999  evthf  31008  stoweidlem34  31362  bnj1400  32991
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