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Theorem cbvrmo 2702
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 2700 . . 3  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
51, 2, 3cbvreu 2701 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
64, 5imbi12i 239 . 2  |-  ( ( E. x  e.  A  ph 
->  E! x  e.  A  ph )  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
7 rmo5 2692 . 2  |-  ( E* x  e.  A  ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
8 rmo5 2692 . 2  |-  ( E* y  e.  A  ps  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
96, 7, 83bitr4i 212 1  |-  ( E* x  e.  A  ph  <->  E* y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1460   E.wrex 2456   E!wreu 2457   E*wrmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rmo 2463
This theorem is referenced by:  cbvrmov  2706  cbvdisj  3987
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