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Theorem rmo2 3076
Description: Alternate definition of restricted "at most one." Note that  E* x  e.  A ph is not equivalent to  E. y  e.  A A. x  e.  A ( ph  ->  x  =  y ) (in analogy to reu6 2954); to see this, let  A be the empty set. However, one direction of this pattern holds; see rmo2i 3077. (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1  |-  F/ y
ph
Assertion
Ref Expression
rmo2  |-  ( E* x  e.  A ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem rmo2
StepHypRef Expression
1 df-rmo 2551 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
2 nfv 1605 . . . 4  |-  F/ y  x  e.  A
3 rmo2.1 . . . 4  |-  F/ y
ph
42, 3nfan 1771 . . 3  |-  F/ y ( x  e.  A  /\  ph )
54mo2 2172 . 2  |-  ( E* x ( x  e.  A  /\  ph )  <->  E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y ) )
6 impexp 433 . . . . 5  |-  ( ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
76albii 1553 . . . 4  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
8 df-ral 2548 . . . 4  |-  ( A. x  e.  A  ( ph  ->  x  =  y )  <->  A. x ( x  e.  A  ->  ( ph  ->  x  =  y ) ) )
97, 8bitr4i 243 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  A. x  e.  A  ( ph  ->  x  =  y ) )
109exbii 1569 . 2  |-  ( E. y A. x ( ( x  e.  A  /\  ph )  ->  x  =  y )  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
111, 5, 103bitri 262 1  |-  ( E* x  e.  A ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531    e. wcel 1684   E*wmo 2144   A.wral 2543   E*wrmo 2546
This theorem is referenced by:  rmo2i  3077  disjiun  4013  rmoanim  27957
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
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-ral 2548  df-rmo 2551
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