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Theorem reusv3i 4217
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
reusv3.2  |-  ( y  =  z  ->  C  =  D )
Assertion
Ref Expression
reusv3i  |-  ( E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
Distinct variable groups:    x, y, z, B    x, C, z   
x, D, y    ph, x, z    ps, x, y
Allowed substitution hints:    ph( y)    ps( z)    A( x, y, z)    C( y)    D( z)

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
2 reusv3.2 . . . . . . 7  |-  ( y  =  z  ->  C  =  D )
32eqeq2d 2093 . . . . . 6  |-  ( y  =  z  ->  (
x  =  C  <->  x  =  D ) )
41, 3imbi12d 232 . . . . 5  |-  ( y  =  z  ->  (
( ph  ->  x  =  C )  <->  ( ps  ->  x  =  D ) ) )
54cbvralv 2578 . . . 4  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  <->  A. z  e.  B  ( ps  ->  x  =  D ) )
65biimpi 118 . . 3  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  A. z  e.  B  ( ps  ->  x  =  D ) )
7 raaanv 3356 . . . 4  |-  ( A. y  e.  B  A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  <->  ( A. y  e.  B  ( ph  ->  x  =  C )  /\  A. z  e.  B  ( ps  ->  x  =  D ) ) )
8 prth 336 . . . . . . 7  |-  ( ( ( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  (
( ph  /\  ps )  ->  ( x  =  C  /\  x  =  D ) ) )
9 eqtr2 2100 . . . . . . 7  |-  ( ( x  =  C  /\  x  =  D )  ->  C  =  D )
108, 9syl6 33 . . . . . 6  |-  ( ( ( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  (
( ph  /\  ps )  ->  C  =  D ) )
1110ralimi 2427 . . . . 5  |-  ( A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
1211ralimi 2427 . . . 4  |-  ( A. y  e.  B  A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
137, 12sylbir 133 . . 3  |-  ( ( A. y  e.  B  ( ph  ->  x  =  C )  /\  A. z  e.  B  ( ps  ->  x  =  D ) )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
146, 13mpdan 412 . 2  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
1514rexlimivw 2474 1  |-  ( E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   A.wral 2349   E.wrex 2350
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-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355
This theorem is referenced by:  reusv3  4218
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