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Theorem cbvrexf 2895
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 1807 . . . 4  |-  F/ y  -.  ph
5 cbvralf.4 . . . . 5  |-  F/ x ps
65nfn 1807 . . . 4  |-  F/ x  -.  ps
7 cbvralf.5 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
87notbid 286 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
91, 2, 4, 6, 8cbvralf 2894 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. y  e.  A  -.  ps )
109notbii 288 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. y  e.  A  -.  ps )
11 dfrex2 2687 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
12 dfrex2 2687 . 2  |-  ( E. y  e.  A  ps  <->  -. 
A. y  e.  A  -.  ps )
1310, 11, 123bitr4i 269 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   F/wnf 1550   F/_wnfc 2535   A.wral 2674   E.wrex 2675
This theorem is referenced by:  cbvrex  2897  reusv2lem4  4694  reusv2  4696  nnwof  10507  dfimafnf  24004  indexa  26333  evth2f  27561  fvelrnbf  27564  evthf  27573  stoweidlem34  27658  bnj1400  28925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680
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