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Theorem mosubop 4575
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 1411 . 2  |-  A. y A. z E* x ph
3 mosubopt 4574 . 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 103   A.wal 1314    = wceq 1316   E.wex 1453   E*wmo 1978   <.cop 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506
This theorem is referenced by:  ovi3  5875  ov6g  5876  oprabex3  5995  axaddf  7644  axmulf  7645
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