Theorem mosubop 2794

Description: "At most one" remains true inside ordered pair quantification.

Hypothesis

Ref	Expression
mosubop.1	|- E*xph

Assertion

Ref	Expression
mosubop	|- E*xE.yE.z(A = <.y, z>. /\ ph)

Distinct variable group:   x,y,z,A

Proof of Theorem mosubop

Step	Hyp	Ref	Expression
1		mosubop.1	. . 3 |- E*xph
2	1	gen2 980	. 2 |- A.yA.zE*xph
3		mosubopt 2793	. 2 |- (A.yA.zE*xph -> E*xE.yE.z(A = <.y, z>. /\ ph))
4	2, 3	ax-mp 7	1 |- E*xE.yE.z(A = <.y, z>. /\ ph)

Syntax hints:   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E*wmo 1374  <.cop 2401

This theorem is referenced by:  oprabex3 4007  oprabval3 4015  oprabval6g 4017  axaddopr 5237  axmulopr 5238

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406

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