Theorem mosubop 2805

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 983	. 2 |- A.yA.zE*xph
3		mosubopt 2804	. 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 954   = wceq 956  E.wex 980  E*wmo 1381  <.cop 2411

This theorem is referenced by:  oprabex3 4022  oprabval3 4030  oprabval6g 4032  axaddopr 5265  axmulopr 5266

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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