Theorem euop2 2801

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 2777	. 2 |- <.A, y>. e. V
2		moop2 2796	. 2 |- E*y x = <.A, y>.
3	1, 2	euxfr2 1922	1 |- (E!xE.y(x = <.A, y>. /\ ph) <-> E!yph)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954  E.wex 978  E!weu 1378  <.cop 2407

This theorem is referenced by:  aceq5lem1 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

Copyright terms: Public domain