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Theorem cbvrmo 2653
Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
cbvral.1  |-  F/ y
ph
cbvral.2  |-  F/ x ps
cbvral.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvrmo  |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvrmo
StepHypRef Expression
1 cbvral.1 . . . 4  |-  F/ y
ph
2 cbvral.2 . . . 4  |-  F/ x ps
3 cbvral.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvrex 2651 . . 3  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
51, 2, 3cbvreu 2652 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
64, 5imbi12i 238 . 2  |-  ( ( E. x  e.  A  ph 
->  E! x  e.  A  ph )  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
7 rmo5 2646 . 2  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
8 rmo5 2646 . 2  |-  ( E* y  e.  A  ps  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
96, 7, 83bitr4i 211 1  |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1436   E.wrex 2417   E!wreu 2418   E*wrmo 2419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-rmo 2424
This theorem is referenced by:  cbvrmov  2657  cbvdisj  3916
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