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Theorem mosubop 4710
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
Hypothesis
Ref Expression
mosubop.1  |-  E* x ph
Assertion
Ref Expression
mosubop  |-  E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph )
Distinct variable group:    x, y, z, A
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem mosubop
StepHypRef Expression
1 mosubop.1 . . 3  |-  E* x ph
21gen2 1461 . 2  |-  A. y A. z E* x ph
3 mosubopt 4709 . 2  |-  ( A. y A. z E* x ph  ->  E* x E. y E. z ( A  =  <. y ,  z
>.  /\  ph ) )
42, 3ax-mp 5 1  |-  E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 104   A.wal 1362    = wceq 1364   E.wex 1503   E*wmo 2039   <.cop 3610
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616
This theorem is referenced by:  ovi3  6034  ov6g  6035  oprabex3  6155  axaddf  7898  axmulf  7899
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