Theorem euop2 2806

Description: Transfer existential uniqueness to second member of an ordered pair.

Assertion

Ref	Expression
euop2	|- (E!xE.y(x = <.A, y>. /\ ph) <-> E!yph)

Distinct variable groups:   ph,x   x,A   x,y

Proof of Theorem euop2

Step	Hyp	Ref	Expression
1		opex 2782	. 2 |- <.A, y>. e. V
2		moop2 2801	. 2 |- E*y x = <.A, y>.
3	1, 2	euxfr2 1926	1 |- (E!xE.y(x = <.A, y>. /\ ph) <-> E!yph)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956  E.wex 980  E!weu 1380  <.cop 2411

This theorem is referenced by:  aceq5lem1 4735

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-nul 2710  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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