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Theorem rmo4f 2935
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmo4f.1  |-  F/_ x A
rmo4f.2  |-  F/_ y A
rmo4f.3  |-  F/ x ps
rmo4f.4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rmo4f  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x, y)

Proof of Theorem rmo4f
StepHypRef Expression
1 rmo4f.1 . . 3  |-  F/_ x A
2 rmo4f.2 . . 3  |-  F/_ y A
3 nfv 1528 . . 3  |-  F/ y
ph
41, 2, 3rmo3f 2934 . 2  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
5 rmo4f.3 . . . . . 6  |-  F/ x ps
6 rmo4f.4 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
75, 6sbie 1791 . . . . 5  |-  ( [ y  /  x ] ph 
<->  ps )
87anbi2i 457 . . . 4  |-  ( (
ph  /\  [ y  /  x ] ph )  <->  (
ph  /\  ps )
)
98imbi1i 238 . . 3  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  <->  ( ( ph  /\  ps )  ->  x  =  y )
)
1092ralbii 2485 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y )  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
114, 10bitri 184 1  |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  (
( ph  /\  ps )  ->  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   F/wnf 1460   [wsb 1762   F/_wnfc 2306   A.wral 2455   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-ral 2460  df-rmo 2463
This theorem is referenced by:  disjxp1  6236
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