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Theorem cbvrexf 2929
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 1812 . . . 4  |-  F/ y  -.  ph
5 cbvralf.4 . . . . 5  |-  F/ x ps
65nfn 1812 . . . 4  |-  F/ x  -.  ps
7 cbvralf.5 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
87notbid 287 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
91, 2, 4, 6, 8cbvralf 2928 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. y  e.  A  -.  ps )
109notbii 289 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. y  e.  A  -.  ps )
11 dfrex2 2720 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
12 dfrex2 2720 . 2  |-  ( E. y  e.  A  ps  <->  -. 
A. y  e.  A  -.  ps )
1310, 11, 123bitr4i 270 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   F/wnf 1554   F/_wnfc 2561   A.wral 2707   E.wrex 2708
This theorem is referenced by:  cbvrex  2931  reusv2lem4  4730  reusv2  4732  nnwof  10548  dfimafnf  24048  indexa  26449  evth2f  27676  fvelrnbf  27679  evthf  27688  stoweidlem34  27773  bnj1400  29281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713
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