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Theorem mosubop 4735
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 1624 . 2  |-  A. y A. z E* x ph
3 mosubopt 4734 . 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 setvar class
Syntax hints:    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617   E*wmo 2285   <.cop 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023
This theorem is referenced by:  ov3  6412  ov6g  6413  oprabex3  6762  axaddf  9511  axmulf  9512
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