Theorem mosubop 4973

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

Step	Hyp	Ref	Expression
1		mosubop.1	. . 3 |- E* x ph
2	1	gen2 1723	. 2 |- A. y A. z E* x ph
3		mosubopt 4972	. 2 |- ( A. y A. z E* x ph -> E* x E. y E. z ( A = <. y , z >. /\ ph ) )
4	2, 3	ax-mp 5	1 |- E* x E. y E. z ( A = <. y , z >. /\ ph )

Syntax hints:   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   E*wmo 2471   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  ov3  6797  ov6g  6798  oprabex3  7157  axaddf  9966  axmulf  9967