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Theorem mosubop 4675
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 1443 . 2  |-  A. y A. z E* x ph
3 mosubopt 4674 . 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 1346    = wceq 1348   E.wex 1485   E*wmo 2020   <.cop 3584
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590
This theorem is referenced by:  ovi3  5986  ov6g  5987  oprabex3  6105  axaddf  7817  axmulf  7818
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