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Theorem cbvrexf 2759
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 1765 . . . 4  |-  F/ y  -.  ph
5 cbvralf.4 . . . . 5  |-  F/ x ps
65nfn 1765 . . . 4  |-  F/ x  -.  ps
7 cbvralf.5 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
87notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
91, 2, 4, 6, 8cbvralf 2758 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. y  e.  A  -.  ps )
109notbii 287 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. y  e.  A  -.  ps )
11 dfrex2 2556 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
12 dfrex2 2556 . 2  |-  ( E. y  e.  A  ps  <->  -. 
A. y  e.  A  -.  ps )
1310, 11, 123bitr4i 268 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623   F/_wnfc 2406   A.wral 2543   E.wrex 2544
This theorem is referenced by:  cbvrex  2761  reusv2lem4  4538  reusv2  4540  nnwof  10285  dfimafnf  23041  indexa  26412  evth2f  27686  fvelrnbf  27689  evthf  27698  stoweidlem34  27783  bnj1400  28868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
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