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Theorem cbvrmo 2763
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 2761 . . 3  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  ps )
51, 2, 3cbvreu 2762 . . 3  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
64, 5imbi12i 316 . 2  |-  ( ( E. x  e.  A  ph 
->  E! x  e.  A  ph )  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
7 rmo5 2756 . 2  |-  ( E* x  e.  A ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
8 rmo5 2756 . 2  |-  ( E* y  e.  A ps  <->  ( E. y  e.  A  ps  ->  E! y  e.  A  ps ) )
96, 7, 83bitr4i 268 1  |-  ( E* x  e.  A ph  <->  E* y  e.  A ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  cbvrmov  2767  cbvdisj  4003
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-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551
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