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Theorem mosubop 4746
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 1602 . 2  |-  A. y A. z E* x ph
3 mosubopt 4745 . 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 369   A.wal 1377    = wceq 1379   E.wex 1596   E*wmo 2276   <.cop 4033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034
This theorem is referenced by:  ov3  6421  ov6g  6422  oprabex3  6770  axaddf  9518  axmulf  9519
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