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Theorem cbvrmo 2577
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 2575 . . 3  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
51, 2, 3cbvreu 2576 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
64, 5imbi12i 237 . 2  |-  ( ( E. x  e.  A  ph 
->  E! x  e.  A  ph )  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
7 rmo5 2570 . 2  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
8 rmo5 2570 . 2  |-  ( E* y  e.  A  ps  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
96, 7, 83bitr4i 210 1  |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1390   E.wrex 2350   E!wreu 2351   E*wrmo 2352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-reu 2356  df-rmo 2357
This theorem is referenced by:  cbvrmov  2581  cbvdisj  3778
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